Universidade Federal de Santa Catarina Curso de Pós-Graduação em Matemática Pura e Aplicada

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛

❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

P✉r❛ ❡ ❆♣❧✐❝❛❞❛

  

❍♦♣❢ ❆❧❣❡❜ró✐❞❡s

❘✐❝❛r❞♦ ❉❛✈✐❞ ▼♦r❛✐s ❉❛ ❙✐❧✈❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳➦ ❉r✳ ❊❧✐❡③❡r ❇❛t✐st❛

  ❋❧♦r✐❛♥ó♣♦❧✐s ❋❡✈❡r❡✐r♦ ❞❡ ✷✵✶✻

  ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛❞❛

  ❍♦♣❢ ❆❧❣❡❜ró✐❞❡s

  ❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈✉rs♦ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐✲ ❝❛❞❛✱ ❞♦ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❋ís✐❝❛s ❡ ▼❛t❡♠át✐❝❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛✱ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✱ ❝♦♠ ➪r❡❛ ❞❡

  ❈♦♥❝❡♥tr❛çã♦ ❡♠ ➪❧❣❡❜r❛✳ ❘✐❝❛r❞♦ ❉❛✈✐❞ ▼♦r❛✐s ❞❛ ❙✐❧✈❛

  ❋❧♦r✐❛♥ó♣♦❧✐s ❋❡✈❡r❡✐r♦ ❞❡ ✷✵✶✻

  

❍♦♣❢ ❆❧❣❡❜ró✐❞❡s

  ♣♦r

  ✶

  ❘✐❝❛r❞♦ ❉❛✈✐❞ ▼♦r❛✐s ❞❛ ❙✐❧✈❛ ❊st❛ ❉✐ss❡rt❛çã♦ ❢♦✐ ❥✉❧❣❛❞❛ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ✏▼❡str❡✑✱

  ➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦ ❡♠ ➪❧❣❡❜r❛✱ ❡ ❛♣r♦✈❛❞❛ ❡♠ s✉❛ ❢♦r♠❛ ✜♥❛❧ ♣❡❧♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡

  ❆♣❧✐❝❛❞❛✳ Pr♦❢✳ ❉r✳ ❉❛♥✐❡❧ ●♦♥ç❛❧✈❡s

  ❈♦♦r❞❡♥❛❞♦r ❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛

  Pr♦❢✳➟ ❉r✳ ❊❧✐❡③❡r ❇❛t✐st❛ ✭❖r✐❡♥t❛❞♦r ✲ ❯❋❙❈✮

  Pr♦❢✳ ❉r✳ ▼❛r❝❡❧♦ ▼✉♥✐③ ❙✐❧✈❛ ❆❧✈❡s ✭❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ P❛r❛♥á ✲ ❯❋P❘✮

  Pr♦❢❛✳ ❉r❛✳ ❆❧❞❛ ❉❛②❛♥❛ ▼❛tt♦s ▼♦rt❛r✐ ✭❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛✲ ❯❋❙❈✮

  Pr♦❢✳ ❉r✳ ❋❡❧✐♣❡ ▲♦♣❡s ❈❛str♦ ✭❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛✲ ❯❋❙❈✮

  ❋❧♦r✐❛♥ó♣♦❧✐s✱ ❋❡✈❡r❡✐r♦ ❞❡ ✷✵✶✻✳

  ✶ ❇♦❧s✐st❛ ❞♦ ❈♦♥s❡❧❤♦ ◆❛❝✐♦♥❛❧ ❞❡ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❈✐❡♥tí✜❝♦ ❡ ❚❡❝♥♦❧ó❣✐❝♦ ✲ ❈◆Pq

  ❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ à ❉❡✉s ♣♦r t♦❞❛ s✉❛ ❜♦♥❞❛❞❡✳ ❆❣r❛❞❡ç♦ à t♦❞❛ ♠✐♥❤❛ ❢❛♠í❧✐❛ ♣♦r s❡♠♣r❡ ❡st❛r ❛♦ ♠❡✉ ❧❛❞♦✱ ♠❡s♠♦ ❡st❛♥❞♦ ❞✐st❛♥t❡✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛✈ós ❏♦sé ❱✐❝❡♥t❡ ❡ ▼❛r✐❛ ●r✐♥❛✉r❛✱ ♣♦r t❡r❡♠ ♣r❛t✐❝❛♠❡♥t❡ ♠❡ ❝r✐❛❞♦✱ à ♠✐♥❤❛ ♠ã❡ ▼❛r✐❧❡♥❡✱ ♣♦r s❡♠♣r❡ ♠❡ ❛♠❛r✱ ♠❡✉ ✐r♠ã♦ ❋❛❜✐❛♥♦ ❡ ♠✐♥❤❛s ✐r♠ãs✱ ❘♦s✐❧❡♥❡✱ ❘♦③✐❧❡✐❞❡ ❡ ❘♦s❡❛♥❡✱ ♣♦r s❡♠♣r❡ ❡st❛r♠♦s ✉♥✐❞♦s ❡♠ t♦❞♦s ♦s ♠♦♠❡♥✲ t♦s✱ s❛✐❜❛♠ q✉❡ ❡✉ ❛♠♦ t♦❞♦s ✈♦❝ês✳ ❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ♥❛♠♦r❛❞❛ ❡ ♠❡✉ ❛♠♦r ❏❛❞♥❛✱ ♣♦r t♦r♥❛r ❛ ♠✐♥❤❛ ✈✐❞❛ ♠❛✐s ❢❡❧✐③ ❡ ♣♦r ♠❡ ♠♦str❛r ♦ ✈❡r❞❛❞❡✐r♦ s❡♥t✐❞♦ ❞♦ ❛♠♦r ❡ ❞❛ ✈✐❞❛✳

  ❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ♣r♦❢❡ss♦r ❊❧✐❡③❡r✱ ♣♦r s❡♠♣r❡ ❛❝r❡❞✐t❛r ♥♦ ♠❡✉ ♣♦t❡♥❝✐❛❧ ❡ ♠❡ ✐♥❝❡♥t✐✈❛r ❛ ❝♦♥t✐♥✉❛r ♠❡✉s ❡st✉❞♦s ❡♠ ▼❛✲ t❡♠át✐❝❛✳ ➚ ♣r♦❢❡ss♦r❛ ❆❧❞❛✱ ♣♦r t✉❞♦ q✉❡ ❢❡③ ♣♦r ♠✐♠✱ ♣♦r t❡r s✐❞♦ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ♥♦ ❚❈❈✱ ♣♦r t❡r ❛❝❡✐t♦ ♣❛rt✐❝✐♣❛r ❞❛ ❜❛♥❝❛ ❞❛ ♠✐✲ ♥❤❛ ❞✐ss❡rt❛çã♦✱ ❢❛③❡♥❞♦ ✈❛❧✐♦s❛s ❝♦rr❡çõ❡s✱ ♣♦r t❛♠❜é♠ ❛❝r❡❞✐t❛r ♥❛ ♠✐♥❤❛ ❝❛♣❛❝✐❞❛❞❡ ❡ ♣❡❧♦s ✈ár✐♦s ❝♦♥s❡❧❤♦s q✉❡ ♠❡ ❞❡✉✱ ♠❡ ❛❥✉❞❛♥❞♦ ❛ ❝r❡s❝❡r ❝♦♠♦ ❡st✉❞❛♥t❡✳ ❆♦s ❞❡♠❛✐s ♣r♦❢❡ss♦r❡s ❞❛ ❜❛♥❝❛✱ ♣♦r t❡r❡♠ ❛❝❡✐t♦ ♦ ❝♦♥✈✐t❡ ♣❛r❛ ❧❡r❡♠ ♠❡✉ tr❛❜❛❧❤♦ ❡ ♣❡❧❛s ❝♦rr❡çõ❡s ❢❡✐t❛s✳

  ❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ❝♦❧❡❣❛ ❞❡ ♠❡str❛❞♦ ❡ ❛♠✐❣♦ ●❛❜r✐❡❧✱ ♣❡❧❛ ❝♦♠♣❛✲ ♥❤✐❛ ♥♦s ✜♥s ❞❡ t❛r❞❡ ❡ ♣❡❧❛ ❛❥✉❞❛ ♥❡ss❡s ❞♦✐s ❛♥♦s✱ ♣♦✐s s❡♠♣r❡ ❡st❡✈❡ ❞✐s♣♦st♦ ❛ t✐r❛r ♠✐♥❤❛s ❞ú✈✐❞❛s✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛♦s ❞❡♠❛✐s ❝♦❧❡❣❛s ❡ ❛♠✐❣♦s q✉❡ ❡♥❝♦♥tr❡✐ ♥❡ss❛ ❥♦r♥❛❞❛✳

  P♦r ✜♠✱ ❛❣r❛❞❡ç♦ ❛♦ ❈◆Pq✱ ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦ q✉❡ ♣♦ss✐❜✐❧✐t♦✉ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳

  ❘❡s✉♠♦

  ❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ tr❛❜❛❧❤♦ é ❞❡✜♥✐r ❡ ❡①❡♠♣❧✐✜❝❛r ❍♦♣❢ ❛❧✲ ❣❡❜ró✐❞❡s✱ q✉❡ sã♦ ✉♠❛ ❞❛s ❣❡♥❡r❛❧✐③❛çõ❡s ❞❡ á❧❣❡❜r❛s ❞❡ ❍♦♣❢✱ s♦❜r❡ ✉♠❛ á❧❣❡❜r❛ ❜❛s❡ ♥ã♦ ❝♦♠✉t❛t✐✈❛✱ ✐st♦ é✱ q✉❡ sã♦ ❝♦♥str✉í❞♦s ❛ ♣❛rt✐r ❞❡ ❜✐♠ó❞✉❧♦s s♦❜r❡ ✉♠ ❛♥❡❧ R✱ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❝♦♠✉t❛t✐✈♦✳ P❛r❛ t❛♥t♦✱ ❞❡✜♥✐♠♦s ❡ ❡①❡♠♣❧✐✜❝❛♠♦s t❛♠❜é♠ ❜✐❛❧❣❡❜ró✐❞❡s✱ q✉❡ ❝♦♥st✐✲ t✉❡♠ ❛ ♠❡❧❤♦r ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ❜✐á❧❣❡❜r❛✳ ❊①♣❧♦r❛♠♦s R ❞✐✈❡rs❛s ♥♦çõ❡s ❡q✉✐✈❛❧❡♥t❡s ❛ ❞❡ ❜✐❛❧❣❡❜ró✐❞❡✱ ❝♦♠♦ ❛s × ✲❜✐á❧❣❡❜r❛s ❞❡ ❚❛❦❡✉❝❤✐✳ ◆♦ ❞❡❝♦rr❡r ❞♦ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❛ t❡♦r✐❛ ❞❡ ❜✐á❧❣❡❜r❛s ❡ ❞❡ á❧❣❡❜r❛s ❞❡ ❍♦♣❢ q✉❡ sã♦ ❡st❡♥❞✐❞♦s ♣❛r❛ ♦ â♠❜✐t♦ ❞❡ ❜✐❛❧❣❡❜ró✐❞❡s ❡ ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s✳ ❈♦♠♦ ✉♠ ❡①❡♠♣❧♦ ✐♠✲ ♣♦rt❛♥t❡ ❞❡ r❡s✉❧t❛❞♦✱ ♣♦❞❡♠♦s ❝✐t❛r ♦ ❢❛t♦ ❞❡ ❛ ❝❛t❡❣♦r✐❛ ❞❡ ♠ó❞✉❧♦s s♦❜r❡ ✉♠ ❜✐❛❧❣❡❜ró✐❞❡ B s❡r ♠♦♥♦✐❞❛❧✱ t❛❧ q✉❡ ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ M M B −→ R R

  é ❡str✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧ ❡ ✉♠ ❛♥á❧♦❣♦ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ❝♦♠ó❞✉❧♦s s♦❜r❡ ❜✐❛❧❣❡❜ró✐❞❡s✳ ◆♦ ✜♥❛❧ ❞♦ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ❛ R ♥♦çã♦ ❞❡ × ✲❍♦♣❢ á❧❣❡❜r❛ ♣r♦♣♦st❛ ♣♦r P✳ ❙❝❤❛✉❡♥❜✉r❣✱ q✉❡ é ✉♠❛ ♥♦çã♦ ♠❛✐s ❣❡r❛❧ ❞♦ q✉❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s✳

  ❆❜str❛❝t

  ❚❤❡ ♠❛✐♥ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s ✇♦r❦ ✐s t♦ ❞❡✜♥❡ ❛♥❞ ❡①❡♠♣❧✐❢② ❍♦♣❢ ❛❧❣❡❜r♦✐❞s✱ ✇❤✐❝❤ ❛r❡ ♦♥❡ ♦❢ t❤❡ ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ ❍♦♣❢ ❛❧❣❡❜r❛s ♦✈❡r ❛♥ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❜❛s✐s✱ t❤❛t ✐s✱ ✇❤✐❝❤ ❛r❡ ❝♦♥str✉❝t❡❞ ❢r♦♠ ❜✐♠♦❞✉✲ ❧❡s ♦✈❡r ❛ r✐♥❣ R✱ ♥♦t ♥❡❝❡ss❛r✐❧② ❝♦♠♠✉t❛t✐✈❡✳ ❚♦ t❤✐s ❡♥❞✱ ✇❡ ❛❧s♦ ❞❡✜♥❡ ❛♥❞ ❡①❡♠♣❧✐❢② ❜✐❛❧❣❡❜r♦✐❞s t❤❛t ❛r❡ t❤❡ ❜❡st ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❝♦♥❝❡♣t ♦❢ ❜✐❛❧❣❡❜r❛✳ ❲❡ ❡①♣❧♦r❡ ❛ ♥✉♠❜❡r ♦❢ ♥♦t✐♦♥s ❡q✉✐✈❛❧❡♥t R t♦ ❜✐❛❧❣❡❜r♦✐❞✱ ❛s ❚❛❦❡✉❝❤✐✬s × ✲❜✐❛❧❣❡❜r❛s✳ ❆❧♦♥❣ t❤✐s ✇♦r❦✱ ✇❡ ♣r❡✲ s❡♥t s♦♠❡ r❡s✉❧ts ♦❢ t❤❡ t❤❡♦r② ♦❢ ❜✐❛❧❣❡❜r❛s ❛♥❞ ❍♦♣❢ ❛❧❣❡❜r❛s ✇❤✐❝❤ ❛r❡ ❡①t❡♥❞❡❞ t♦ t❤❡ s❝♦♣❡ ♦❢ ❜✐❛❧❣❡❜r♦✐❞s ❛♥❞ ❍♦♣❢ ❛❧❣❡❜r♦✐❞s✳ ❆s ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛♥ ✐♠♣♦rt❛♥t r❡s✉❧t✱ ✇❡ ❝❛♥ ♠❡♥t✐♦♥ t❤❡ ❢❛❝t t❤❛t t❤❡ ❝❛t❡✲ ❣♦r② ♦❢ ♠♦❞✉❧❡s ♦✈❡r ❛ ❜✐❛❧❣❡❜r♦✐❞ B ✐s ♠♦♥♦✐❞❛❧✱ s✉❝❤ t❤❛t ❢♦r❣❡tt✐♥❣ B −→ R R M ❢✉♥❝t♦r M ✐s str✐❝t❧② ♠♦♥♦✐❞❛❧ ❛♥❞ ❛♥❛❧♦❣♦✉s t♦ t❤❡ ❝❛s❡ ♦❢ ❝♦♠♦❞✉❧❡s ♦✈❡r ❜✐❛❧❣❡❜r♦✐❞s✳ ❆t t❤❡ ❡♥❞ ♦❢ t❤❡ ✇♦r❦ ✇❡ ♣r❡s❡♥t t❤❡ R ♥♦t✐♦♥ ♦❢ ❛ × ✲❍♦♣❢ ❛❧❣❡❜r❛✱ ♣r♦♣♦s❡❞ ❜② P✳ ❙❝❤❛✉❡♥❜✉r❣✱ ✇❤✐❝❤ ✐s ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❝♦♥❝❡♣t t❤❛♥ ♦❢ ❍♦♣❢ ❛❧❣❡❜r♦✐❞s✳

  ❮♥❞✐❝❡

  ■♥tr♦❞✉çã♦ ①✈

  ✶ ❈❛t❡❣♦r✐❛s ▼♦♥♦✐❞❛✐s ✹ ✶✳✶ ❉❡✜♥✐çã♦ ❡ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❋✉♥t♦r❡s ▼♦♥♦✐❞❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸ ❘✲❛♥é✐s ✭♠♦♥ó✐❞❡s✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

  ✶✳✸✳✶ ▼ó❞✉❧♦s s♦❜r❡ ▼♦♥ó✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✹ ❘✲❝♦❛♥é✐s ✭❝♦♠♦♥ó✐❞❡s✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

  ✶✳✹✳✶ ❈♦♠ó❞✉❧♦s ❙♦❜r❡ ❈♦♠♦♥ó✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✺ ❉✉❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

  ✷ ❇✐❛❧❣❡❜ró✐❞❡s ✹✽

  ✷✳✶ ❉❡✜♥✐çã♦ ❡ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✷✳✷ ❉✉❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✷✳✸ ❈♦♥str✉çõ❡s ❞❡ ◆♦✈♦s ❇✐❛❧❣❡❜ró✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶

  ✷✳✸✳✶ ❚✇✐st ❞❡ ❉r✐♥❢❡❧❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✷✳✸✳✷ ❚✇✐st ♣♦r ✷✲❝♦❝✐❝❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✷✳✸✳✸ ❉✉❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✷✳✸✳✹ ❇✐❛❧❣❡❜ró✐❞❡ ❞❡ ❈♦♥♥❡s✲▼♦s❝♦✈✐❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷ ✷✳✸✳✺ ❊①t❡♥sã♦ ❊s❝❛❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✼

  ✷✳✹ ❆ ❈❛t❡❣♦r✐❛ ▼♦♥♦✐❞❛❧ ❞❡ ▼ó❞✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✸ ✷✳✺ ❆ ❈❛t❡❣♦r✐❛ ▼♦♥♦✐❞❛❧ ❞❡ ❈♦♠ó❞✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾ ✷✳✻ ❱❡rsõ❡s ❊q✉✐✈❛❧❡♥t❡s ❞❛ ❉❡✜♥✐çã♦ ❞❡ ❇✐❛❧❣❡❜ró✐❞❡ ✳ ✳ ✳ ✶✷✻

  ✸ ❍♦♣❢ ❆❧❣❡❜ró✐❞❡s ✶✸✻ ✸✳✶ ❉❡✜♥✐çã♦ ❡ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✻ ✸✳✷ Pr♦♣r✐❡❞❛❞❡s ❇ás✐❝❛s ❞❡ ❍♦♣❢ ❆❧❣❡❜ró✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✾ ✸✳✸ ◆♦çõ❡s ❆❧t❡r♥❛t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✼

  ✸✳✸✳✶ ❍♦♣❢ ❆❧❣❡❜ró✐❞❡s ❞❡ ▲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✼ R ✸✳✸✳✷ × ✲❍♦♣❢ ➪❧❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✾

  ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✶✼✸ ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✶✼✹

  ■♥tr♦❞✉çã♦

  ❖ q✉❡ é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡❄ ❘❡s♣♦♥❞❡♥❞♦ ❛ ❡ss❛ ♣❡r❣✉♥t❛ ❡♠ ✉♠❛ ❢r❛s❡✱ ❞✐r❡♠♦s q✉❡ ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ q✉❡ s❡ ❝♦♥❤❡❝❡ ❝♦♠♦ á❧❣❡❜r❛ ❞❡ ❍♦♣❢✱ s♦❜r❡ ✉♠❛ á❧❣❡❜r❛ ❜❛s❡ ♥ã♦ ❝♦♠✉t❛t✐✈❛✳ ❖✉ s❡❥❛✱ ♦ ♠❡❧❤♦r ❡①❡♠♣❧♦ ❞❡ ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ é ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ❝♦♥✈✐♥❝❡♥t❡ ❞❡ r❡s✉❧t❛❞♦s ♥❛ t❡♦r✐❛ ❞❡ á❧❣❡❜r❛s ❞❡ ❍♦♣❢ sã♦ ❡st❡♥❞✐❞♦s ♣❛r❛ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s✳ ❖✉ ♠❡❧❤♦r ❛✐♥❞❛✱ ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ❛té ❡♥tã♦ ♥ã♦ r❡s♦❧✈✐❞♦s ❝♦♠ á❧❣❡❜r❛s ❞❡ ❍♦♣❢✱ sã♦ r❡s♦❧✈✐❞♦s ♥♦ â♠❜✐t♦ ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s✳ ❉❡ss❛ ❢♦r♠❛✱ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s ❢♦r♥❡❝❡♠✲♥♦s r❡s✉❧t❛❞♦s ❞❡ ❛♠❜♦s ♦s t✐♣♦s✱ ❛q✉❡❧❡s q✉❡ s❡ ❡st❡♥❞❡♠ ❞❡ á❧❣❡❜r❛s ❞❡ ❍♦♣❢ ❡ t❛♠❜é♠ ❛q✉❡❧❡s q✉❡ sã♦ ❝♦♥❝❡✐✲ t✉❛❧♠❡♥t❡ ♥♦✈♦s✳

  ❖r✐❣✐♥❛❧♠❡♥t❡✱ ♥♦ ❝❛♠♣♦ ❞❛ t♦♣♦❧♦❣✐❛ ❛❧❣é❜r✐❝❛✱ ♦ t❡r♠♦ ✬❍♦♣❢ ❛❧❣❡✲ ❜ró✐❞❡✬ ❢♦✐ ✉s❛❞♦ ♣♦r ❉♦✉❣❧❛s ❈✳ ❘❛✈❡♥❡❧ ❡♠❬✷✹❪ ♣❛r❛ ❞❡s❝r❡✈❡r ♦❜❥❡t♦s ❝♦❣r✉♣ó✐❞❡s ♥❛ ❝❛t❡❣♦r✐❛ ❞❡ á❧❣❡❜r❛s ❝♦♠✉t❛t✐✈❛s✳ ❊st❡s sã♦ ❡①❡♠♣❧♦s ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s ❝♦♠ ❡str✉t✉r❛ ❞❡ á❧❣❡❜r❛ s✉❜❥❛❝❡♥t❡ ❝♦♠✉t❛t✐✈❛✳ ❚❛♠❜é♠ ❡♠ ❬✷✵❪✱ ❛✐♥❞❛ ♥❛ ár❡❛ ❞❛ t♦♣♦❧♦❣✐❛ ❛❧❣é❜r✐❝❛✱ ❡♥❝♦♥tr❛✲s❡ ✉♠❛ ❛♣❧✐❝❛çã♦✲❡①❡♠♣❧♦ ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳ ❊♠ ❬✷✶❪✱ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s ♥ã♦ ❝♦♠✉t❛t✐✈♦s tê♠ s✐❞♦ ✉s❛❞♦s✱ ♠❛s ❛✐♥❞❛ s♦❜r❡ á❧❣❡❜r❛s ❜❛s❡ ❝♦✲ ♠✉t❛t✐✈❛s✱ ❝♦♠♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❞❡ ✉♠ ❡st✉❞♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞♦s ❢❡✐①❡s ❞❡ ✜❜r❛❞♦s ♣r✐♥❝✐♣❛✐s ❝♦♠ s✐♠❡tr✐❛s ❞❡ ❣r✉♣ó✐❞❡s✳ ❆❧é♠ ❞❛ t♦♣♦❧♦✲ ❣✐❛ ❛❧❣é❜r✐❝❛✱ ♣♦❞❡♠♦s ❝✐t❛r ♦✉tr❛s ár❡❛s ❡♠ q✉❡ ❢♦r❛♠ ❛♣❧✐❝❛❞♦s ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s✱ ❝♦♠♦ ❣❡♦♠❡tr✐❛ ❞❡ P♦✐ss♦♥ ❡ t♦♣♦❧♦❣✐❛✳

  ❊♠ ✶✾✾✺ ♠♦t✐✈❛❞❛ ♣❡❧❛ ♥♦çã♦ ❞❡ ❣r✉♣ó✐❞❡s ❞❡ P♦✐ss♦♥✱ ❡♠ ❣❡♦♠❡✲ tr✐❛ ❞❡ P♦✐ss♦♥✱ ❏✳ ❍✳ ▲✉ ❞❡✜♥✐✉ ❡♠ ❬✶✽❪ ✉♠❛ ♥♦çã♦ ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ ❡♠ q✉❡ ❛ á❧❣❡❜r❛ s✉❜❥❛❝❡♥t❡ ♥ã♦ ♣r❡❝✐s❛✈❛ s❡r ❝♦♠✉t❛t✐✈❛✳ ❆ ❞❡✜♥✐çã♦ ❡♥✈♦❧✈❡ ❛ ♥♦çã♦ ❞❡ ❜✐✲❛❧❣❡❜ró✐❞❡ ❝♦♠ ❛♥tí♣♦❞❛ ❜✐❥❡t✐✈❛✱ q✉❡ é s♦❜r❡❝❛r✲ r❡❣❛❞❛ ❝♦♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✉♠❛ s❡çã♦ ♣❛r❛ ♦ ❡♣✐♠♦r✜s♠♦ ❝❛♥ô♥✐❝♦ A⊗ k A −→ A⊗ L A

  ✱ ❡♠ q✉❡ A ❡ L sã♦ á❧❣❡❜r❛s s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✉♥✐❞❛❞❡ k✳

  ❊♠ ❬✶✹❪ ▲✳ ❑❛❞✐s♦♥ ❡ ❑✳ ❙③❧❛❝❤á♥②✐ ❣❡♥❡r❛❧✐③❛r❛♠ ❛ ♥♦çã♦ ❞❡ ❜✐✲

  ❛❧❣❡❜ró✐❞❡ ❞❛❞❛ ♣♦r ▲✉ ❡♠ ❬✶✽❪✱ ♣❛r❛ ❜✐❛❧❣❡❜ró✐❞❡s à ❡sq✉❡r❞❛ ❡ à ❞✐r❡✐t❛✳ ◆♦ ❛rt✐❣♦ ❬✸❪ ♠♦t✐✈❛❞♦s ♣❡❧♦ ❡st✉❞♦ ❞❡ ❡①t❡♥sõ❡s ❞❡ ❋r♦❜❡♥✐✉s ❞❡ ♣r♦❢✉♥❞✐❞❛❞❡ ✷ ❡♠ ❬✷❪✱ ●✳ ❇¨o❤♠ ❡ ❑✳ ❙③❧❛❝❤á♥②✐ ✐♥tr♦❞✉③✐r❛♠ ✉♠❛ ♥♦✈❛ ♥♦çã♦ ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳ ❚❛❧ ♥♦çã♦ é ❞❛❞❛ ♣❡❧❛ ❛♥tí♣♦❞❛ ❜✐✲ ❥❡t✐✈❛✱ ♠❛s q✉❡ ❝♦♥❡❝t❛ ❛s ❞✉❛s ❡st✉t✉r❛s ❞❡ ❜✐❛❧❣❡❜ró✐❞❡s à ❡sq✉❡r❞❛ ❡ à ❞✐r❡✐t❛✱ s❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞♦ ❡♣✐♠♦r✜s♠♦ ❝❛♥ô♥✐❝♦ ❝✐t❛❞♦ ❛❝✐♠❛✳ ❆ ♣r♦♣♦st❛ ❞❡ ✉♠❛ ❛♥tí♣♦❞❛ é ❜❛s❡❛❞❛ ❡♠ ✉♠❛ s✐♠♣❧❡s ♦❜s❡r✈❛çã♦✳ ❆ ❛♥tí♣♦❞❛ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢ H é ✉♠ ♠♦r✜s♠♦ ❞❡ ❜✐á❧❣❡❜r❛s op H −→ H cop ✳ ❊♠ ✷✵✵✸ ♥♦ ❛rt✐❣♦ ❬✹❪ ●✳ ❇¨o❤♠ ❞❡✜♥✐✉ ❛ ♥♦çã♦ ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✱ ❡st✉❞❛❞❛ ♥❡ss❡ tr❛❜❛❧❤♦✱ s❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❛ ❛♥tí♣♦❞❛ s❡r ❜✐❥❡t✐✈❛✳

  ❆♣r❡s❡♥t❡♠♦s ✉♠❛ ❞✐s♣♦s✐çã♦ ❣❡r❛❧ ❞❡ ♥♦ss♦ tr❛❜❛❧❤♦✱ q✉❡ é ❞✐✲ ✈✐❞♦ ❡♠ três ♣❛rt❡s✳ ◆♦ ❝❛♣ít✉❧♦ ✶ ❞❡✜♥✐♠♦s ❡ ❡①❡♠♣❧✐✜❝❛♠♦s ❛ ♥♦çã♦ ❞❡ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ❚❛❧ ♥♦çã♦ ♥♦s ♣❡r♠✐t❡ ❣❡♥❡r❛❧✐③❛r á❧❣❡❜r❛s ❡ ❝♦á❧❣❡❜r❛s✱ ❛ss✐♠ ❝♦♠♦ ♠ó❞✉❧♦s ❡ ❝♦♠ó❞✉❧♦s s♦❜r❡ t❛✐s ♦❜❥❡t♦s✱ r❡s♣❡❝✲ t✐✈❛♠❡♥t❡✳ ❆♣❡s❛r ❞❡ s❡r ✉♠ ❝♦♥❝❡✐t♦ ❣❡r❛❧✱ ♥♦s r❡str✐♥❣✐♠♦s ❛♣❡♥❛s R R M

  ❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❞♦s R✲❜✐♠ó❞✉❧♦s✱ ❡♠ q✉❡ R é ✉♠❛ á❧❣❡❜r❛ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✉♥✐❞❛❞❡ k✳ ◆♦ ✜♥❛❧ ❞♦ ❝❛♣ít✉❧♦ ❡st✉❞❛✲ R R M ♠♦s ✉♠ ♣♦✉❝♦ ❞❛ ❞✉❛❧✐❞❛❞❡ ❡♥tr❡ R✲❛♥é✐s ✭♦❜❥❡t♦s á❧❣❡❜r❛s ❡♠ ✮ R R M ❡ R✲❝♦❛♥é✐s ✭♦❜❥❡t♦s ❝♦á❧❣❡❜r❛s ❡♠ ✮✳

  ◆♦ ❝❛♣ít✉❧♦ ✷ ❡st✉❞❛♠♦s ❛ ♥♦çã♦ ❞❡ ❜✐❛❧❣❡❜ró✐❞❡s s❡❣✉♥❞♦ ❬✼❪✳ ❇✐✲ R ❛❧❣❡❜ró✐❞❡✱ ✐♥✈❡♥t❛❞♦ ♣♦r ❚❛❦❡✉❝❤✐ ❝♦♠♦ × ✲❜✐á❧❣❡❜r❛ ❡♠ ❬✷✽❪✱ é ✉♠❛ ❞❛s ❣❡♥❡r❛❧✐③❛çõ❡s ❞❡ ❜✐á❧❣❡❜r❛s s♦❜r❡ ✉♠❛ á❧❣❡❜r❛ ❜❛s❡ ♥ã♦ ❝♦♠✉t❛✲ t✐✈❛✳ ▼❛s ♦ q✉❡ s✐❣♥✐✜❝❛ ❛ á❧❣❡❜r❛ ❜❛s❡ R ❞❡ ✉♠ ❜✐❛❧❣❡❜ró✐❞❡ s❡r ♥ã♦ ❝♦♠✉t❛t✐✈❛❄ ▲❡♠❜r❡♠♦s q✉❡ ✉♠❛ ❜✐á❧❣❡❜r❛ é ✉♠ k✲♠ó❞✉❧♦✱ ❝♦♠ ❡str✉✲ t✉r❛s ❝♦♠♣❛tí✈❡✐s ❞❡ á❧❣❡❜r❛ ❡ ❝♦á❧❣❡❜r❛✳ P♦r ❛♥❛❧♦❣✐❛✱ ❡♠ ✉♠ ❜✐❛❧❣❡✲ ❜ró✐❞❡ ❛ ❡str✉t✉r❛ ❞❡ ❝♦á❧❣❡❜r❛ é s✉❜st✐t✉í❞❛ ♣♦r ✉♠ ❝♦❛♥❡❧ s♦❜r❡ ✉♠❛ k

  ✲á❧❣❡❜r❛ R✱ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❝♦♠✉t❛t✐✈❛✳ ❚❛♠❜é♠ ❛ ❡str✉t✉r❛ ❞❡ á❧❣❡❜r❛ é s✉❜st✐t✉í❞❛ ♣♦r ✉♠ ❛♥❡❧ s♦❜r❡ ✉♠❛ á❧❣❡❜r❛ ❜❛s❡ ♥ã♦ ❝♦♠✉t❛✲ t✐✈❛✳ ◆♦ ❡♥t❛♥t♦✱ ♣❛r❛ ❢♦r♠✉❧❛r ❛ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❡♥tr❡ ❛s ❡str✉t✉r❛s op

  ❞❡ ❛♥❡❧ ❡ ❝♦❛♥❡❧✱ ❛ á❧❣❡❜r❛ ❜❛s❡ ❞♦ ❛♥❡❧ ♥ã♦ é R✱ ♠❛s R ⊗ R ✳ ❍á ✉♠ ❝♦♥s❡♥s♦ ♥❛ ❧✐t❡r❛t✉r❛ ❞❡ q✉❡ ❜✐❛❧❣❡❜ró✐❞❡ é ❛ ♠❡❧❤♦r ❣❡♥❡r❛❧✐③❛çã♦ ❞❡ ❜✐á❧❣❡❜r❛ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ✉♠ ❛♥❡❧ ❜❛s❡ ♥ã♦ ❝♦♠✉t❛t✐✈♦✳

  ❈♦♠❡ç❛♠♦s ♦ ❝❛♣ít✉❧♦ ✷ ❞❡✜♥✐♥❞♦ ❡ ❡①❡♠♣❧✐✜❝❛♥❞♦ ❛s ❞✉❛s ❡str✉✲ t✉r❛s ❞❡ ❜✐❛❧❣❡❜ró✐❞❡s✱ à ❡sq✉❡r❞❛ ❡ à ❞✐r❡✐t❛✳ ❉❡♣♦✐s✱ ♠♦str❛♠♦s q✉❡✱ ❛♦ ❝♦♥trár✐♦ ❞❡ ❜✐á❧❣❡❜r❛s ❝✉❥♦s ❛①✐♦♠❛s sã♦ ❛✉t♦✲❞✉❛✐s✱ ♦s ❛①✐♦♠❛s ❞❡ ❜✐❛❧❣❡❜ró✐❞❡s ♥ã♦ sã♦ ❛✉t♦✲❞✉❛✐s ♥♦ ♠❡s♠♦ s❡♥t✐❞♦✱ ♠❛s ❝♦♠ ❛ ❤✐♣ót❡s❡ ❛❞✐❝✐♦♥❛❧ ❞❡ s❡r ♣r♦❥❡t✐✈♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱ ♦ ❞✉❛❧ ❞❡ ✉♠ ❜✐❛❧❣❡❜ró✐❞❡

  ∗

  B é t❛♠❜é♠ ✉♠ ❜✐❛❧❣❡❜ró✐❞❡✳ ❆ s❛❜❡r✱ ♦ ❞✉❛❧ à ❡sq✉❡r❞❛ ❞❡ ✉♠ R✲ ❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛ B é ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❊♠ s❡❣✉✐❞❛✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ❝♦♥str✉çõ❡s ❞❡ ♥♦✈♦s ❜✐❛❧❣❡❜ró✐❞❡s ❛ ♣❛rt✐r ❞❡

  ♦✉tr♦s ❞❛❞♦s✱ ❝♦♠♦✿ t✇✐st ❞❡ ❉r✐♥❢❡❧❞✱ ❞✉♣❧♦ ❝♦❝✐❝❧♦ t✇✐st✱ ❜✐❛❧❣❡❜ró✐❞❡ ❞❡ ❈♦♥♥❡s✲▼♦s❝♦✈✐❝✐ ❡ ❡①t❡♥sã♦ ❞❡ ❡s❝❛❧❛r❡s✳

  ❈♦♥t✐♥✉❛♥❞♦✱ ❛✐♥❞❛ ♥♦ ❝❛♣ít✉❧♦ ✷✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❛♥á❧♦❣♦ ♣❛r❛ ❜✐❛❧❣❡❜ró✐❞❡s ❞♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧✿ ❯♠❛ k✲á❧❣❡❜r❛ B é B ✉♠❛ ❜✐á❧❣❡❜r❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ❝❛t❡❣♦r✐❛ M ❞♦s B✲♠ó❞✉❧♦s à ❞✐✲ B −→ M k r❡✐t❛✱ é ♠♦♥♦✐❞❛❧ t❛❧ q✉❡ ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ M é ❡str✐✲ t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧✱ ✈❡r ❬✷✸❪✳ ❚❛♠❜é♠ ♠♦str❛♠♦s q✉❡ ❝♦♥t✐♥✉❛ s❡♥❞♦ ✈á❧✐❞♦ ♣❛r❛ ❜✐❛❧❣❡❜ró✐❞❡s ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✿ ❆ ❝❛t❡❣♦r✐❛ ❞♦s ❝♦♠ó❞✉✲ ❧♦s à ❞✐r❡✐t❛ s♦❜r❡ B✱ ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ♠♦♥♦✐❞❛❧✱ t❛❧ q✉❡ ♦ ❢✉♥t♦r k ❡sq✉❡❝✐♠❡♥t♦ ♣❛r❛ M é ❡str✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧✳ ◆♦ ✜♥❛❧ ❞♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❜✐❛❧❣❡❜ró✐❞❡s à ❡sq✉❡r❞❛ s❡❣✉♥❞♦ ●✳ R ❇¨o❤♠ ❡♠ ❬✼❪✱ ❜✐❛❧❣❡❜ró✐❞❡s ❝♦♠ â♥❝♦r❛ ❡♠ ❬✸✶❪ ❡ × ✲❜✐á❧❣❡❜r❛s ❡♠ ❬✷✽❪✳ ❚❛❧ ❡q✉✐✈❛❧ê♥❝✐❛ ❢♦✐ ♠♦str❛❞❛ ♣♦r ❇r③❡③✐♥s❦✐ ❡ ▼✐❧✐t❛r✉ ❡♠ ❬✽❪✳

  ❋✐♥❛❧♠❡♥t❡✱ ♥♦ ❝❛♣ít✉❧♦ ✸ ❝♦♠❡ç❛♠♦s ❞❡✜♥✐♥❞♦ ❡ ❡①❡♠♣❧✐✜❝❛♥❞♦ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s s❡❣✉♥❞♦ ❬✼❪✳ ❊♠ s❡❣✉✐❞❛✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ♣r♦✲ ♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s ❝♦♠♦✿ P❛r❛ ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ op op H

  = (H L , H R , S) R ) , (H L ) , S) ✱ ❛ tr✐♣❧❛ ((H cop cop é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ op op s♦❜r❡ á❧❣❡❜r❛s ❜❛s❡s R ❡ L ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡❀ ❙❡ ❛ ❛♥tí♣♦❞❛ S é op op R ) , (H L ) , S )

−1

  ❜✐❥❡t✐✈❛✱ ❡♥tã♦ ((H é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ s♦❜r❡ á❧❣❡✲ L ) cop , (H R ) cop , S ) −1 ❜r❛s ❜❛s❡s R ❡ L✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ t❛♠❜é♠✱ ((H op op é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ s♦❜r❡ á❧❣❡❜r❛s ❜❛s❡s L ❡ R ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❚❛♠❜é♠✱ ♠♦str❛♠♦s q✉❡ ❛ ❛♥tí♣♦❞❛ é ✉♠ ♠♦r✜s♠♦ ❞❡ ❜✐á❧❣❡❜ró✐❞❡s R ) −→ H L . op à ❡sq✉❡r❞❛ S : (H cop ◆♦ ✜♥❛❧ ❞♦ ❝❛♣ít✉❧♦ ✸ ❛♣r❡s❡♥t❛✲ ♠♦s ❞✉❛s ♥♦çõ❡s ❛❧t❡r♥❛t✐✈❛s ❞❡ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢✳ ❈♦♠♣❛r❛♥❞♦ ❛ ♥♦çã♦ ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ s❡❣✉♥❞♦ ●✳ ❇ö❤♠ ❬✼❪ ❡ ❛ ♥♦✲ çã♦ ✐♥tr♦❞✉③✐❞❛ ♣♦r ❏✳ ❍✳ ▲✉ ❬✶✽❪✱ ❝♦♥st❛t❛♠♦s q✉❡ ♥❡♠ ✉♠❛ ❞❛s ❞✉❛s é ♠❛✐s ❣❡r❛❧ q✉❡ ❛ ♦✉tr❛✳ ▼❛s q✉❡ ❛ ♥♦çã♦ ❛♣r❡s❡♥t❛❞❛ ♥❡st❡ tr❛❜❛❧❤♦ R s❡❣✉♥❞♦ ●✳ ❇¨o❤♠ ♣❡rt❡♥❝❡ ❛ ❝❧❛ss❡ ❞❡ × ✲❍♦♣❢ á❧❣❡❜r❛s ♣r♦♣♦st❛ ♣♦r P✳ ❙❝❤❛✉❡♥❜✉r❣ ❡♠ ❬✷✺❪✳

  ❈♦♠♦ ♣ré✲r❡q✉✐s✐t♦s ♣❛r❛ ❛ ❧❡✐t✉r❛ ❞♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✱ s✉❣❡r✐♠♦s ❛♦ ❧❡✐t♦r t❡r ♥♦çã♦ ❞❛ t❡♦r✐❛ ❜ás✐❝❛ ❞❡ á❧❣❡❜r❛s ❞❡ ❍♦♣❢ ✈❡r ❬✶✶❪ ❡ ❬✶✻❪✳ ❚❛♠❜é♠ s✉❣❡r✐♠♦s ✉♠ ♣♦✉❝♦ ❞❡ ❡st✉❞♦ s♦❜r❡ ❛ t❡♦r✐❛ ❞❡ ❝♦❛♥é✐s ❡ ❝♦♠ó❞✉❧♦s ✈❡r ❬✾❪✳ ❆♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦✱ ❝✐t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s út❡✐s ❡ s✉❛s r❡❢❡rê♥❝✐❛s✱ à ♠❡❞✐❞❛ q✉❡ ✐ss♦ ❢♦r ♥❡❝❡ssár✐♦✳

  ❈❛♣ít✉❧♦ ✶ ❈❛t❡❣♦r✐❛s ▼♦♥♦✐❞❛✐s

  ◆❡st❡ ❝❛♣ít✉❧♦✱ ❞❡✜♥✐♠♦s ❡ ❡①❡♠♣❧✐✜❝❛♠♦s ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s✱ q✉❡ sã♦ ❝❛t❡❣♦r✐❛s q✉❡ ♣❡r♠✐t❡♠ ❞❡✜♥✐r♠♦s ❣❡♥❡r❛❧✐③❛çõ❡s ❞❡ ♦❜❥❡t♦s ❛❧❣é❜r✐❝♦s ❝♦♠♦✿ k✲á❧❣❡❜r❛s ❡ k✲❝♦á❧❣❡❜r❛s✱ ❛ss✐♠ ❝♦♠♦✱ r❡s♣❡❝t✐✈❛✲ ♠❡♥t❡✱ ♠ó❞✉❧♦s ❡ ❝♦♠ó❞✉❧♦s s♦❜r❡ t❛✐s ♦❜❥❡t♦s✱ ❡♠ q✉❡ k é ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✉♥✐❞❛❞❡✳

  ▼❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ s❡ R é ✉♠❛ á❧❣❡❜r❛ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛✲ t✐✈♦ k✱ ✈❛♠♦s ❡st✉❞❛r R✲❛♥é✐s ❡ R✲❝♦❛♥é✐s✱ q✉❡ ❝♦♠♦ ✈❛♠♦s ✈❡r ♠❛✐s ❛❞✐❛♥t❡✱ sã♦ ♦❜❥❡t♦s á❧❣❡❜r❛s ❡ ♦❜❥❡t♦s ❝♦á❧❣❡❜r❛s ♥❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐✲ ❞❛❧ ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ◆♦ ✜♥❛❧ ❞♦ ❝❛♣ít✉❧♦✱ ❡st✉❞❛♠♦s ✉♠ ♣♦✉❝♦ s♦❜r❡ ❞✉❛❧✐❞❛❞❡ ❡♥tr❡ t❛✐s ♦❜❥❡t♦s✳ P❛r❛ ❡st❡ ❝❛♣ít✉❧♦ s❡❣✉✐♠♦s ❬✶❪✱ ❬✾❪ ❡ ❬✶✾❪ ❝♦♠♦ r❡❢❡rê♥❝✐❛✳

  ✶✳✶ ❉❡✜♥✐çã♦ ❡ ❊①❡♠♣❧♦s

  ❉❡✜♥✐çã♦ ✶✳✶ ❯♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ é ✉♠❛ sê①t✉♣❧❛ (C, ⊗, ✶, a, l, r) ✱ ❡♠ q✉❡ C é ✉♠❛ ❝❛t❡❣♦r✐❛✱ ⊗ : C × C −→ C é ✉♠ ❢✉♥t♦r✱ ❝❤❛♠❛❞♦

  ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❡ ✶ é ✉♠ ♦❜❥❡t♦ ❡♠ C✱ ❝❤❛♠❛❞♦ ♦❜❥❡t♦ ✉♥✐❞❛❞❡✳ ❆❧é♠ ❞✐ss♦✱ a é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ❡♥tr❡ ♦s ❢✉♥t♦r❡s (− ⊗ −) ⊗ − ❡ − ⊗ (− ⊗ −) ❞❡ C × C × C ♣❛r❛ C✱ l é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ❡♥tr❡ ♦s ❢✉♥t♦r❡s ✶ ⊗ − ❡ Id ❞❡ C ♣❛r❛ C ❡ r é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ❡♥tr❡ ♦s ❢✉♥t♦r❡s − ⊗ ✶ ❡ Id ❞❡ C ♣❛r❛ C✱ t❛✐s q✉❡ ♣❛r❛ q✉❛✐sq✉❡r ♦❜❥❡t♦s

  A, B, C ❡ D ∈ C ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s a ⊗D A⊗B,C,D ABC ((A ⊗B) ⊗C) ⊗ D a uu )) (A ⊗(B ⊗C)) ⊗D (A ⊗B)⊗(C ⊗D) a A,B⊗C,D a ABC⊗D

  A⊗((B ⊗C) ⊗D) // A⊗(B ⊗(C ⊗D)), A

  ⊗a BCD

a

A, ✶,B

  // (A ⊗ A ⊗ (

  ✶) ⊗ B ✶ ⊗ B) r A ⊗l B

A

⊗B '' A ⊗ B, sã♦ ❝♦♠✉t❛t✐✈♦s✳

  ❖ ❢❛t♦ ❞❡ a s❡r ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ♥♦s ❞✐③ q✉❡ ♣❛r❛ ❝❛❞❛ tr✐♣❧❛ ❞❡ ♦❜❥❡t♦s A, B, C ❡♠ C✱ t❡♠♦s ✉♠ ✐s♦♠♦r✜s♠♦ a A,B,C : (A ⊗ B) ⊗ C −→ A ⊗ (B ⊗ C), q✉❡ s❛t✐s❢❛③ ❞❡t❡r♠✐♥❛❞♦ ❞✐❛❣r❛♠❛✱ ✈❡r ✭❬✷✾❪✱ ♣á❣✳✷✹✮✳ ❉❡ ♠❛♥❡✐r❛ A : A : A ⊗ ❛♥á❧♦❣❛✱ t❡♠♦s ♦s ✐s♦♠♦r✜s♠♦s l ✶ ⊗ A −→ A ❡ r ✶ −→ A✳

  ◆❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ ♦ ❢❛t♦ ❞❡ q✉❡ ♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛ é ❝❤❛♠❛❞♦ ❛①✐♦♠❛ ❞♦ ♣❡♥tá❣♦♥♦ ❡ ♦ ❢❛t♦ ❞♦ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛ ❝♦♠✉✲ t❛r é ❝❤❛♠❛❞♦ ❛①✐♦♠❛ ❞♦ tr✐â♥❣✉❧♦✳ ❊ss❡s ❛①✐♦♠❛s ❡①♣r❡ss❛♠ q✉❡ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♦❜❥❡t♦s ❡stá ❜❡♠ ❞❡✜♥✐❞♦✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ♣♦s✐çã♦ ❞♦s ♣❛rê♥t❡s❡s ❡ q✉❡ ✶ é ✉♠❛ ✉♥✐❞❛❞❡ ♣❛r❛ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧✳ P❡r❝❡❜❛ ❛✐♥❞❛ ♥❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ q✉❡ ❞❡♥♦t❛♠♦s ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ✉♠ ♦❜❥❡t♦ ❡ ♦ ♣ró♣r✐♦ ♦❜❥❡t♦ ♣❡❧♦ ♠❡s♠♦ sí♠❜♦❧♦✳ ❱❛✲ ♠♦s ❞❡♥♦t❛r ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ✉♠ ♦❜❥❡t♦ ❞❡st❛ ❢♦r♠❛✱ s❡♠♣r❡ q✉❡ ♥ã♦ ❤♦✉✈❡r ❛♠❜✐❣✉✐❞❛❞❡✳ ❉❡✜♥✐çã♦ ✶✳✷ ❯♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ (C, ⊗, ✶, a, l, r) é ❞✐t❛ s❡r ❡s✲ tr✐t❛ s❡ a, l, r sã♦ ❛s ✐❞❡♥t✐❞❛❞❡s ♥♦s r❡s♣❡❝t✐✈♦s ♦❜❥❡t♦s✳ ❖✉ s❡❥❛✱ ♣❛r❛ q✉❛✐sq✉❡r A, B ❡ C ∈ C t❡♠♦s

  (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C) ❡ ✶ ⊗ A = A = A ⊗ ✶. P♦r s✐♠♣❧✐❝✐❞❛❞❡✱ ✈❛♠♦s ♥♦s r❡❢❡r✐r ❛ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❝♦♠♦

  (C, ⊗, ✶) ♦✉ ❛♣❡♥❛s C✱ ❛♦ ✐♥✈és ❞❡ (C, ⊗, ✶, a, l, r)✳

  ❊①❡♠♣❧♦ ✶✳✸ ❆ ❝❛t❡❣♦r✐❛ Set ❞♦s ❝♦♥❥✉♥t♦s é ♠♦♥♦✐❞❛❧✳ ❖ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ é ♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ × : Set × Set −→ Set✳ ❖ ♦❜❥❡t♦ ✉♥✐❞❛❞❡ ✶ = {∗} é ✉♠ ❝♦♥❥✉♥t♦ ✉♥✐tár✐♦ q✉❛❧q✉❡r✳ P❛r❛ q✉❛✐sq✉❡r X, Y, Z ∈ Set

  ❞❡✜♥✐♠♦s a XY Z : (X × Y ) × Z → X × (Y × Z); ((x, y), z) 7→ (x, (y, z)) l X : {∗} × X → X;

  (∗, x) 7→ x r X : X × {∗} → X.

  (x, ∗) 7→ x ▼♦str❡♠♦s q✉❡ (Set, ×, {∗}, a, l, r) é ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ❉❡

  ❢❛t♦✱ ❝❧❛r♦ q✉❡ a✱ l ❡ r sã♦ ❜✐❥❡çõ❡s✱ ♦✉ s❡❥❛✱ ✐s♦♠♦r✜s♠♦s ❡♠ Set✳ ▲♦❣♦✱ ❜❛st❛ ♠♦str❛r♠♦s ❛ ❝♦♥❞✐çã♦ ❞❛ ❞❡✜♥✐çã♦ ❞❡ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳

  ′ ′ ′

  , B , C P❛r❛ t❛♥t♦✱ s❡❥❛♠ A, B, C, A ❝♦♥❥✉♥t♦s q✉❛✐sq✉❡r ❡♠ Set ❡ f :

  ′ ′ ′

  A −→ A ✱ g : B −→ B ✱ h : C −→ C ❢✉♥çõ❡s q✉❛✐sq✉❡r ❡♠ Set✳

  ❱❡r✐✜q✉❡♠♦s q✉❡ ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s ❝♦♠✉t❛♠ a A,B,C //

  (A × B) × C A × (B × C) f

  

(f ×g)×h ×(g×h)

′ ′ ′ ′ ′ ′

  (A × B ) × C // A × (B × C ), l r A A a A′ ,B′ ,C′ // //

  {∗} × A A A × {∗} A f f f

  {∗}×f ×{∗} ′ ′ ′ ′

  {∗} × A // A A × {∗} // A . l A′ r A′ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a ∈ A✱ b ∈ B ❡ c ∈ C✱ t❡♠♦s

  (f × (g × h)) ◦ a A,B,C ((a, b), c) = (f × (g × h))(a, (b, c)) = (f (a), (g × h)(b, c)) = (f (a), (g(b), h(c))) ′ ′ ′ = a A ,B ,C ((f (a), g(b)), h(c)) ′ ′ ′ = a A ,B ,C ((f × g)(a, b), h(c)) ′ ′ ′ = a A ,B ,C ◦ ((f × g) × h)((a, b), c),

  ′ ′

  l A (({∗} × f )(∗, a)) = l A (∗, f (a)) = f (a) = f (l A (∗, a)) = f ◦ l A (∗, a) ❡ ′ ′ r A ((f × {∗})(a, ∗)) = r A (f (a), ∗) = f (a) = f (r A (a, ∗)) = f ◦ r A (a, ∗).

  ▲♦❣♦✱ a✱ l ❡ r sã♦ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s✳ ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ✈❛❧❡♠ ♦s ❛①✐♦♠❛s ❞♦ ♣❡♥tá❣♦♥♦ ❡ ❞♦ tr✐â♥❣✉❧♦✳ ❙❡❥❛♠ A, B, C, D ❝♦♥❥✉♥t♦s q✉❛✐sq✉❡r ❡♠ Set✱ a ∈ A✱ b ∈ B✱ c ∈ C ❡ d ∈ D q✉❛✐sq✉❡r✳ ❚❡♠♦s

  (A × a B,C,D ) ◦ a A,B ×C,D ◦ (a A,B,C × D)(((a, b), c), d) B,C,D A,B = (A × a ) ◦ a ×C,D ((a, (b, c)), d) = (A × a B,C,D )(a, ((b, c), d)) = (a, (b, (c, d))),

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s a A,B,C ◦ a A (((a, b), c), d) = (a A,B,C × D)((a, b), (c, d))

  ×D ×B,C,D

  = (a, (b, (c, d))), t❛♠❜é♠ t❡♠♦s (A × l B ) ◦ a A, ((a, ∗), b) = (A × l B )(a, (∗, b))

  {∗},B

  = (a, l B (∗, b)) = (a, b) = (r A (a, ∗), b) = (r A × B)((a, ∗), b).

  P♦rt❛♥t♦✱ (Set, ×, {∗}, a, l, r) é ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ k ❊①❡♠♣❧♦ ✶✳✹ ❆ ❝❛t❡❣♦r✐❛ V ect ❞♦s ❡s♣❛ç♦s ✈❡t♦r✐❛s s♦❜r❡ ✉♠ ❝♦r♣♦ k k × V ect k −→ V ect k

  é ♠♦♥♦✐❞❛❧✳ ❉❡✜♥✐♠♦s ⊗ : V ect ❝♦♠♦ ♦ ♣r♦❞✉t♦ k t❡♥s♦r✐❛❧ s♦❜r❡ ♦ ❝♦r♣♦ ❜❛s❡ ⊗ ❡ ✶ := k✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ q✉❛✐sq✉❡r V, U k

  ❡ W ∈ V ect ❡ λ ∈ V ✱ ❞❡✜♥✐♠♦s a V U W : (V ⊗ k U ) ⊗ k W → V ⊗ k (U ⊗ k W ) (v ⊗ u) ⊗ w 7→ v ⊗ (u ⊗ w); l V : k ⊗ V →

  V λ ⊗ v 7→ λv; r V : V ⊗ k →

  V v ⊗ λ 7→ λv.

  ❊①❡♠♣❧♦ ✶✳✺ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❆ ❝❛t❡❣♦r✐❛ Supervect ❞❡ s✉♣❡r✲ k ❡s♣❛ç♦s ✈❡t♦r✐❛✐s é ♠♦♥♦✐❞❛❧✳ ❖s ♦❜❥❡t♦s ❡♠ Supervect sã♦ k✲❡s♣❛ç♦s k

  ⊕ V ✈❡t♦r✐❛✐s✱ ❡q✉✐♣❛❞♦s ❞❡ ✉♠❛ ❣r❛❞✉❛çã♦ s♦❜r❡ Z

  2 ✱ ♦✉ s❡❥❛✱ V = V 1 ✳

  ❖s ♠♦r✜s♠♦s sã♦ tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s q✉❡ ♣r❡s❡r✈❛♠ ❛ ❣r❛❞✉❛çã♦✳ k W ❙❡ V, W ∈ Supervect ❞❡✜♥✐♠♦s V ⊗ W := V ⊗ ✱ ❝♦♠ ❣r❛❞✉❛çã♦ k (V ⊗ W ) = V ⊗ k W ⊕ V ⊗ k W , (V ⊗ W ) = V ⊗ k W ⊕ V ⊗ k W .

  1

  1

  1

  1

  1

  ❉❡✜♥✐♠♦s t❛♠❜é♠ a, l ❡ r ❝♦♠♦ ♥♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ❞❛ ❝❛t❡❣♦r✐❛ vect k ✳ R R M ❊①❡♠♣❧♦ ✶✳✻ ❙❡❥❛♠ k ✉♠ ❝♦r♣♦ ❡ R ✉♠❛ k✲á❧❣❡❜r❛✳ ❆ ❝❛t❡❣♦r✐❛ ❞♦s R✲❜✐♠ó❞✉❧♦s é ♠♦♥♦✐❞❛❧✳ ❉❡✜♥✐♠♦s

  M M M ⊗ : R R × R R −→ R R R

  ❝♦♠♦ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❜❛❧❛♥❝❡❛❞♦ ♣♦r R✱ ❞❡♥♦t❛❞♦ ♣♦r ⊗ ✳ ❖ ♦❜❥❡t♦ R R M ✉♥✐❞❛❞❡ é ❛ k✲á❧❣❡❜r❛ R✳ P❛r❛ q✉❛✐sq✉❡r M, N ❡ P ∈ ✱ m ∈ M✱ n ∈ N

  ✱ p ∈ P ❡ b ∈ R✱ ❞❡✜♥✐♠♦s a, l ❡ r✱ ♣♦r a M N P : (M ⊗ R N ) ⊗ R P → M ⊗ R (N ⊗ R P ) (m ⊗ n) ⊗ p 7→ m ⊗ (n ⊗ p); l M : R ⊗ M → M b ⊗ m 7→ b · m; r M : M ⊗ R → M m ⊗ b 7→ m · b. ❊①❡♠♣❧♦ ✶✳✼ ❙❡❥❛ (H, µ, η, ∆, ε) ✉♠❛ ❜✐á❧❣❡❜r❛ s♦❜r❡ k ✉♠ ❝♦r♣♦✳ ❆ H M H

  ❝❛t❡❣♦r✐❛ ❞♦s H✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ é ♠♦♥♦✐❞❛❧✳ ❉❡✜♥✐♠♦s ⊗ : M M M

  × H −→ H k ❝♦♠♦ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ s♦❜r❡ ♦ ❝♦r♣♦ ❞❡ ❜❛s❡ ⊗ ✳ H M

  ❙❡ M, N ∈ ❛ ❡str✉t✉r❛ ❞❡ H✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ ❡♠ M ⊗N é ❞❛❞❛✱ ♣❛r❛ t♦❞♦ m ∈ M n ∈ N ❡ h ∈ H✱ ♣♦r h · (m ⊗ n) = h · m ⊗ h · n,

  (1) (2)

  ⊗ k h ❡♠ q✉❡ ∆(h) = h (1) (2) ✳ ❖s ♠♦r✜s♠♦s ♥❛t✉r❛✐s a, l, r sã♦ ♦s ♠❡s♠♦s ❞♦ ❡①❡♠♣❧♦ ❞❛ ❝❛t❡❣♦r✐❛ V ect k ✳ ❖ ♦❜❥❡t♦ ✉♥✐❞❛❞❡ é ♦ ❝♦r♣♦ k ❝♦♠ ❛çã♦ ❞❡ H ❞❛❞❛ ♣❡❧❛ ❝♦✉♥✐❞❛❞❡✱ ♦✉ s❡❥❛✱ ♣❛r❛ q✉❛✐sq✉❡r h ∈ H ❡ λ ∈ k t❡♠♦s h · λ = ε(h)λ✳ Pr✐♠❡✐r❛♠❡♥t❡ ✈❛♠♦s ✈❡r q✉❡ M ⊗ N é H✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛✱ ❝♦♠ ❡str✉t✉r❛ ❞❛❞❛ ❛❝✐♠❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M✱ n ∈ N ❡

  h, k ∈ H t❡♠♦s h · (k · (m ⊗ n)) = h · ((k · m) ⊗ (k · n))

  (1) (2)

  = (h · (k · m)) ⊗ (h · (k · n))

  (1) (1) (2) (2)

  = (h k · m) ⊗ (h k · n)

  (1) (1) (2) (2)

  = ((hk) · m) ⊗ ((hk) · n)

  (1) (2) = (hk) · (m ⊗ n).

  ❆❣♦r❛ ✈❛♠♦s ✈❡r q✉❡ k é H✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛✱ ❝♦♠ ❡str✉t✉r❛ ❞❛❞❛ ❛❝✐♠❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r λ ∈ k ❡ h, k ∈ H t❡♠♦s h · (k · λ) = h · (ε(k)λ)

  = ε(h)ε(k)λ = ε(hk)λ M N P : (M ⊗ N ) ⊗ P −→ M ⊗ (N ⊗ P ) = (hk) · λ.

  ❱❛♠♦s ✈❡r✐✜❝❛r q✉❡ a é ♠♦r✜s♠♦ H M ❞❡ H✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r M, N, P ∈ ✱ m ∈ M

  ✱ n ∈ N✱ p ∈ P ❡ h ∈ H✱ t❡♠♦s a M N P (h · ((m ⊗ n) ⊗ p)) = a M N P (h · (m ⊗ n) ⊗ h · p)

  (1) (2)

  = a M N P ((h · m ⊗ h · n) ⊗ h · p)

  (1)(1) (1)(2) (2)

  = a M N P ((h · m ⊗ h · n) ⊗ h · p)

  (1) (2)(1) (2)(2)

  = h · m ⊗ (h · n ⊗ h · p)

  (1) (2)(1) (2)(2)

  = h · m ⊗ (h · (n ⊗ p))

  (1) (2)

  = h · (m ⊗ (n ⊗ p)) M : k ⊗ M −→ M = h · a M N P ((m ⊗ n) ⊗ p). ❚❛♠❜é♠ t❡♠♦s q✉❡ l é ♠♦r✜s♠♦ ❞❡ H✲♠ó❞✉❧♦s à H M ❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r M ∈ ✱ λ ∈ k ❡ h ∈ H✱ t❡♠♦s l M (h · (λ ⊗ m)) = l M (h · λ ⊗ h · m)

  (1) (2)

  = l M (ε(h )λ ⊗ h · m)

  (1) (2)

  = ε(h )λ(h · m)

  (1) (2)

  = λε(h )h · m

  (1) (2)

  = λh · m = h · (λm) = h · l M (λ ⊗ m). M : M ⊗k −→ M

  ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ♠♦str❛✲s❡ q✉❡ r é ♠♦r✜s♠♦ ❞❡ H✲ ♠ó❞✉❧♦ à ❡sq✉❡r❞❛✳ ◆ã♦ é ❞✐❢í❝✐❧ ♠♦str❛r q✉❡ a, l ❡ r sã♦ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s ❡ q✉❡ ♦s ❛①✐♦♠❛s ❞♦ ♣❡♥tá❣♦♥♦ ❡ ❞♦ tr✐â♥❣✉❧♦ sã♦ s❛t✐s❢❡✐t♦s✳

  ❊①❡♠♣❧♦ ✶✳✽ ❙❡❥❛ H ✉♠❛ ❜✐á❧❣❡❜r❛ s♦❜r❡ k ✉♠ ❝♦r♣♦✳ ❆ ❝❛t❡❣♦r✐❛ H M

  ❞♦s H✲❝♦♠ó❞✉❧♦s à ❞✐r❡✐t❛ é ♠♦♥♦✐❞❛❧✳ ❉❡✜♥✐♠♦s H H H ⊗ : M × M −→ M k H

  ❝♦♠♦ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ s♦❜r❡ ♦ ❝♦r♣♦ ❞❡ ❜❛s❡ ⊗ ✳ ❙❡ M, N ∈ M ✱ ❛ ❡str✉t✉r❛ ❞❡ H✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❡♠ M ⊗N é ❞❛❞❛✱ ♣❛r❛ t♦❞♦ m ∈ M ❡ n ∈ N✱ ♣♦r

  ρ M N : M ⊗ N → (M ⊗ N ) ⊗ H

  (0) (0) (1) (1) m ⊗ n 7→ m ⊗ n ⊗ m n .

  ❖ ♦❜❥❡t♦ ✉♥✐❞❛❞❡ é ♦ ❝♦r♣♦ k ❝♦♠ ❝♦❛çã♦ ❞❡ H ❞❛❞❛✱ ♣❛r❛ t♦❞♦ λ ∈ k✱ k : k −→ k ⊗ H k (λ) = λ ⊗ 1 H ♣♦r ρ ✱ ρ ✳ ❉❡✜♥✐♠♦s a, l, r ❝♦♠♦ ♥♦ ❡①❡♠♣❧♦ ❞❛ ❝❛t❡❣♦r✐❛ vect k ✳ M N P : (M ⊗ N ) ⊗ P −→ M ⊗ (N ⊗ P ) ❱❛♠♦s ✈❡r q✉❡ a é ♠♦r✜s♠♦ H ❞❡ H✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❉❡ ❢❛t♦✱ s❡❥❛♠ M, N ❡ P ∈ M q✉❛✐sq✉❡r✱ ❡♥tã♦ ♣❛r❛ t♦❞♦ m ∈ M✱ n ∈ N ❡ p ∈ P ✱ q✉❡r❡♠♦s ♠♦str❛r q✉❡ (a M N P ⊗ H) ◦ ρ M = ρ M,N ◦ a M N P .

  ⊗N,P ⊗P

  ❉❡ ❢❛t♦✱ (a M N P ⊗ H) ◦ ρ M ⊗N,P ((m ⊗ n) ⊗ p) =

  (0) (0) (1) (1)

  = (a M N P ⊗ H)((m ⊗ n) ⊗ p ⊗ (m ⊗ n) p )

  (0) (0) (0) (1) (1) (1)

  = m ⊗ (n ⊗ p ) ⊗ m n p

  (0) (0) (1) (1)

  = m ⊗ (n ⊗ p) ⊗ m (n ⊗ p) = ρ M,N (m ⊗ (n ⊗ p))

  ⊗P = ρ M,N ◦ a M N P ((m ⊗ n) ⊗ p). ⊗P M : k ⊗ M −→ M M : k ⊗ M −→ M

  ❚❡♠♦s t❛♠❜é♠ q✉❡ l ❡ r sã♦ H ♠♦r✜s♠♦s ❞❡ H✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛❧q✉❡r M ∈ M ❡ ♣❛r❛ t♦❞♦ m ∈ M ❡ λ ∈ k t❡♠♦s

  (0) (1)

  (l M ⊗ H) ◦ ρ k,M (λ ⊗ m) = (l M ⊗ H)(λ ⊗ m ⊗ 1 H m )

  (0) (1)

  = λm ⊗ m = ρ M (λm) = ρ M ◦ l M (λ ⊗ m).

  (0) (1)

  (r M ⊗ H) ◦ ρ M,k (m ⊗ λ) = (r M ⊗ H)(m ⊗ λ ⊗ m

  1 H )

  (0) (1)

  = m λ ⊗ m = ρ M (mλ) = ρ M ◦ l M (m ⊗ λ).

  ◆ã♦ é ❞✐❢í❝✐❧ ♠♦str❛r q✉❡ a, l, r sã♦ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s ❡ q✉❡ ✈❛❧❡♠ ♦s ❛①✐♦♠❛s ❞♦ ♣❡♥tá❣♦♥♦ ❡ ❞♦ tr✐â♥❣✉❧♦✳ ❊①❡♠♣❧♦ ✶✳✾ ❈♦♥s✐❞❡r❡ C ✉♠❛ ❝❛t❡❣♦r✐❛✱ ❛ ❝❛t❡❣♦r✐❛ ❞❡ ❡♥❞♦❢✉♥t♦✲ r❡s End(C) é ♠♦♥♦✐❞❛❧ ❝♦♠ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❞❛❞♦ ♣❡❧❛ ❝♦♠♣♦s✐çã♦ ❞❡ ❢✉♥t♦r❡s✳ ◆❡st❛ ❝❛t❡❣♦r✐❛ ❛ ❝♦♠♣♦s✐çã♦ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s é ❛ ✈❡rt✐❝❛❧✳ ❖ ♦❜❥❡t♦ ✉♥✐❞❛❞❡ é ♦ ❢✉♥t♦r ✐❞❡♥t✐❞❛❞❡✱ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❞❡ ❞✉❛s tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s é ❛ ❝♦♠♣♦s✐çã♦ ❤♦r✐③♦♥t❛❧✱ ♦✉ s❡❥❛✱ s❡

  ′ ′ ′ ′

  F, F , G, G ∈ End(C) −→ G ❡ α : F −→ G✱ β : F tr❛♥s❢♦r♠❛çõ❡s

  ♥❛t✉r❛✐s✱ ❡♥tã♦ ♣❛r❛ t♦❞♦ X ∈ C✱ t❡♠♦s

  ′ ′ X = G(β (α ⊗ β) : F (F (X)) −→ G(G (X)), X ) ◦ α F t❛❧ q✉❡ (α ⊗ β) (X) ✳ ❈❧❛r❛♠❡♥t❡✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞♦

  ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❡♠ End(C)✱ ❡st❡ é ✉♠ ❡①❡♠♣❧♦ ❞❡ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡str✐t❛✳ ❊①❡♠♣❧♦ ✶✳✶✵ ❈♦♥s✐❞❡r❡ (C, ⊗, ✶, a, r, l) ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ❆ rev rev rev rev rev rev rev

  , ⊗ , , a , r , l ) = C ❝❛t❡❣♦r✐❛ (C ✶ é ♠♦♥♦✐❞❛❧✱ ❡♠ q✉❡ C rev rev

  : C × C −→ C Y = ❝♦♠♦ ❝❛t❡❣♦r✐❛✱ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ⊗ ✱ X ⊗ Y ⊗ X

  ✱ ♣❛r❛ t♦❞♦ X, Y ∈ C✳ ❊ t❛♠❜é♠ ♣❛r❛ q✉❛✐sq✉❡r X, Y, Z ∈ C rev −1 rev rev rev = a = l X = r X = t❡♠♦s a ❀ r ❀ l ❡ ✶ ✶✳ X,Y,Z Z,Y,X X X

  ✶✳✷ ❋✉♥t♦r❡s ▼♦♥♦✐❞❛✐s

  ❉❡✜♥✐çã♦ ✶✳✶✶ ❙❡❥❛♠ (C, ⊗, ✶, a, l, r) ❡ (D, ⊠, I, a, l, r) ❞✉❛s ❝❛t❡❣♦✲ r✐❛s ♠♦♥♦✐❞❛✐s✳ ❯♠ ❢✉♥t♦r ♠♦♥♦✐❞❛❧ ❡♥tr❡ C ❡ D é ✉♠❛ tr✐♣❧❛ (F, ϕ , φ)

  : I −→ ❡♠ q✉❡ F : C −→ D é ✉♠ ❢✉♥t♦r ✭❝♦✈❛r✐❛♥t❡✮✱ ϕ

  F ( ✶) ♠♦r✜s♠♦ ❡ φ : (− ⊠ −) ◦ (F, F ) ⇒ F ◦ (− ⊗ −) é ✉♠❛ tr❛♥s❢♦r✲

  ♠❛çã♦ ♥❛t✉r❛❧ ❡♥tr❡ C × C ❡ D t❛❧ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r ♦❜❥❡t♦s A, B ❡ C ∈ C

  ✱ t❡♠♦s l r F (A) F (A) // //

  I ⊠ F (A) F (A) F (A) ⊠ I F (A) ϕ F F F OO OO

  (l ) (A) (r )

⊠F (A) A ⊠φ A

  // F ( // F (A ⊗ F ( F (A) ⊠ F (

  ✶) ⊠ F (A) ✶ ⊗ A), ✶) ✶) φ φ ✶A A ✶

  ❡ F

  (A) B,C ⊠φ

  F (A) ⊠ (F (B) ⊠ F (C)) // F (A) ⊠ F (B ⊗ C) a φ F A,B⊗C (A)F (B)F (C) OO (F (A) ⊠ F (B)) ⊠ F (C) F (A ⊗ (B ⊗ C)) φ F OO AB ⊠F (C) A,B,C (a ) // F ((A ⊗ B) ⊗ C).

  F (A ⊗ B) ⊠ F (C) φ A⊗B,C , φ)

  ❉❡✜♥✐çã♦ ✶✳✶✷ ❯♠ ❢✉♥t♦r ♠♦♥♦✐❞❛❧ (F, ϕ é ❞✐t♦ s❡r ❢♦rt❡ ♦✉ ❢♦rt❡♠❡♥t❡ ♠♦♥♦✐❞❛❧ s❡ ϕ é ✉♠ ✐s♦♠♦r✜s♠♦ ❡ φ é ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳

  , φ) ❉❡✜♥✐çã♦ ✶✳✶✸ ❯♠ ❢✉♥t♦r ♠♦♥♦✐❞❛❧ (F, ϕ é ❞✐t♦ s❡r ❡str✐t♦ ♦✉ ❡str✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧ s❡ ϕ ❡ φ ❢♦r❡♠ ✐❞❡♥t✐❞❛❞❡✳ f

  G ❊①❡♠♣❧♦ ✶✳✶✹ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ✜♥✐t♦✱ k ✉♠ ❝♦r♣♦ ❡ Rep ❛ ❝❛✲ t❡❣♦r✐❛ ❞❛s r❡♣r❡s❡♥t❛çõ❡s k✲❧✐♥❡❛r❡s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ❞♦ ❣r✉♣♦ G✳ ❖s ♦❜❥❡t♦s ♥❡st❛ ❝❛t❡❣♦r✐❛ sã♦ r❡♣r❡s❡♥t❛çõ❡s (V, ρ)✱ ♦✉ s❡❥❛✱ ρ : G → End(V)

  é ♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✳ ▼♦r✜s♠♦s ♥❡st❛ ❝❛t❡❣♦r✐❛ sã♦ ♠♦r✲ ✜s♠♦s ❞❡ r❡♣r❡s❡♥t❛çõ❡s✱ ♦✉ s❡❥❛✱ ❞❛❞❛s ❞✉❛s r❡♣r❡s❡♥t❛çõ❡s (V, ρ) ❡ (W, σ)

  ✱ ✉♠ ♠♦r✜s♠♦ ❡♥tr❡ ❛♠❜❛s é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r f : V → W t❛❧ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r g ∈ G ❡ v ∈ V✱ t❡♠♦s f (ρ(g)(v)) = σ(g)(f (v)). f

  G → V ect ❖ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ U : Rep k é ❡str✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧✳ f

  G ❆ ❝❛t❡❣♦r✐❛ Rep é ♠♦♥♦✐❞❛❧✳ ❖ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❞❡ ❞✉❛s r❡✲

  ♣r❡s❡♥t❛çõ❡s (V, ρ) ⊗ (W, σ) := (V ⊗ W, ρ ⊗ σ) ❡♠ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r g ∈ G ✱ v⊗w t❡♠✲s❡ q✉❡ (ρ⊗σ)(g)(v⊗w) = ρ(g)(v)⊗σ(g)(w)✱ é t❛♠❜é♠ k )

  ✉♠❛ r❡♣r❡s❡♥t❛çã♦✳ ❚❡♠♦s t❛♠❜é♠ q✉❡ (k, ρ é ✉♠❛ r❡♣r❡s❡♥t❛çã♦✱ k (g)(1 k ) = 1 k ❡♠ q✉❡ ρ ♣❛r❛ t♦❞♦ g ∈ G✳ P♦rt❛♥t♦✱ k é ❛ ✉♥✐❞❛❞❡ V,W,U (V ⊗ W) ⊗ U → V ⊗ (W ⊗ U) ♠♦♥♦✐❞❛❧✳ ❖s ♠♦r✜s♠♦s ❝❛♥ô♥✐❝♦s a l V : k ⊗ V → V V : V ⊗ k → V

  ❡ r sã♦ ♠♦r✜s♠♦s ❞❡ r❡♣r❡s❡♥t❛çõ❡s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r❡♣r❡s❡♥t❛çõ❡s (V, ρ)✱ (W, σ)✱ (U, λ) ❡ ♣❛r❛ q✉❛✐sq✉❡r v ∈ V✱ w ∈ W✱ u ∈ U✱ g ∈ G ❡ r ∈ k t❡♠♦s a V,W,U (((ρ ⊗ σ) ⊗ λ)(g)((v ⊗ w) ⊗ u))

  = a V,W,U ((ρ ⊗ σ)(g)(v ⊗ w) ⊗ λ(g)(u)) = a V,W,U ((ρ(g)(v) ⊗ σ(g)(w)) ⊗ λ(g)(u))

  = ρ(g)(v) ⊗ (σ(g)(w) ⊗ λ(g)(u)) = ρ(g)(v) ⊗ ((σ ⊗ λ)(g)(w ⊗ u)) = (ρ ⊗ (σ ⊗ λ))(g)(v ⊗ (w ⊗ u)) = (ρ ⊗ (σ ⊗ λ))(g)(a V,W,U ((v ⊗ w) ⊗ u)), l V ((ρ k ⊗ ρ)(g)(r ⊗ v)) = l V (ρ k (g)(r) ⊗ ρ(g)(v))

  = l V (r ⊗ ρ(g)(v)) = rρ(g)(v) = ρ(g)(rv) = ρ(l V (r ⊗ v))

  ❡ r V ((ρ ⊗ ρ k )(g)(v ⊗ r)) = r V (ρ(g)(v) ⊗ r) = ρ(g)(v)r = ρ(vr) = ρ(r V (v ⊗ r)). ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ ♦s ❛①✐♦♠❛s ❞♦ ♣❡♥tá❣♦♥♦ ❡ ❞♦ tr✐â♥❣✉❧♦ sã♦ s❛t✐s❢❡✐t♦s✳ ❚❛♠❜é♠ é tr✐✈✐❛❧ ❛ ✈❡r✐✜❝❛çã♦ ❞❡ q✉❡ ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ f

  U : Rep G → V ect k é ❡str✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧✳ ▼❛s ♥❡♠ s❡♠♣r❡ ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ é ❡str✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧✳

  ❈♦♠♦ ✈❛♠♦s ✈❡r ♥♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦✳ R R → V ect M ❊①❡♠♣❧♦ ✶✳✶✺ ❙❡❥❛ R ✉♠❛ á❧❣❡❜r❛ s♦❜r❡ ♦ ❝♦r♣♦ k ❡ U : k ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ ❞❛ ❝❛t❡❣♦r✐❛ ❞♦s R✲❜✐♠ó❞✉❧♦s ♣❛r❛ ❛ ❝❛t❡❣♦r✐❛ ❞♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s s♦❜r❡ k✳ ◆❡ss❡ ❝❛s♦ U é ♠♦♥♦✐❞❛❧ ♠❛s ♥ã♦ é ❡str✐t♦✳ R R M

  ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r M, N ∈ ✱ m ∈ M✱ n ∈ N ❡ λ ∈ k ❞❡✜♥✐♠♦s ❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ M,N (m ⊗ k n) = m ⊗ R n φ : − ⊗ k − ⇒ − ⊗ R −, t❛❧ q✉❡ φ ❡ ♦ ♠♦r✜s♠♦s ϕ é ❛ ✐♥❥❡çã♦

  ϕ : k → R.

  λ 7→ λ1 R . ➱ s✐♠♣❧❡s ✈❡r✐✜❝❛r♠♦s q✉❡ ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s ❝♦♠✉t❛♠✳ ❖ q✉❡

  ♠♦str❛ q✉❡ U é ♠♦♥♦✐❞❛❧✱ ♣♦ré♠ ♥ã♦ é ❡str✐t♦✱ ♣♦✐s ❛ tr❛♥s❢♦r♠❛çã♦

  ♥❛t✉r❛❧ φ ❡ ♦ ♠♦r✜s♠♦ ϕ ♥ã♦ sã♦ ✐❞❡♥t✐❞❛❞❡s✳

  ∼ ∼ = =

  // // k ⊗ k M M M ⊗ k k M ϕ l M OO OO r

  

⊗M ⊗ϕ M

M

  R ⊗ k M // R ⊗ R M, M ⊗ k R // M ⊗ R R φ φ R,M M,RM

  

⊗φ

N,P

  M ⊗ k (N ⊗ k P ) // M ⊗ k (N ⊗ R P ) a M,N,P OO φ M,N ⊗RP (M ⊗ k N ) ⊗ k P M ⊗ R (N ⊗ R P ) φ ⊗ P M,N,P M,N k OO a (M ⊗ R N ) ⊗ k P // (M ⊗ R N ) ⊗ R P. φ M ⊗RN,P

  ◆♦s ❞♦✐s ♣r✐♠❡✐r♦s ❞✐❛❣r❛♠❛s ❡st❛♠♦s ❢❛③❡♥❞♦ ✉♠ ❛❜✉s♦ ❞❡ ♥♦t❛çã♦ k M ∼ k k ∼ q✉❛♥❞♦ ❞❡♥♦t❛♠♦s ♦s ✐s♦♠♦r✜s♠♦s ❝❛♥ô♥✐❝♦s k ⊗ = M ❡ M ⊗ = M

  ❝♦♠ ♦ ♠❡s♠♦ sí♠❜♦❧♦✳

  ✶✳✸ ❘✲❛♥é✐s ✭♠♦♥ó✐❞❡s✮

  ❉❡✜♥✐çã♦ ✶✳✶✻ ❯♠ ♦❜❥❡t♦ á❧❣❡❜r❛ ♦✉ ✉♠ ▼♦♥ó✐❞❡ ❡♠ ✉♠❛ ❝❛t❡✲ ❣♦r✐❛ ♠♦♥♦✐❞❛❧ (C, ⊗, ✶) é ✉♠❛ tr✐♣❧❛ (A, µ, η)✳ ❖♥❞❡ A é ✉♠ ♦❜❥❡t♦ ❡♠ C ❡ µ : A ⊗ A −→ A✱ η : ✶ −→ A sã♦ ♠♦r✜s♠♦s ❡♠ C✱ s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ ✉♥✐t❛❧✐❞❛❞❡✱ ♦✉ s❡❥❛✱ ♦s s❡❣✉✐♥t❡s ❞✐✲ ❛❣r❛♠❛s ❝♦♠✉t❛♠ µ ⊗A

  // (A ⊗ A) ⊗ A A ⊗ A a A,A,A µ ✭✶✳✶✮ A ⊗ (A ⊗ A) A

  ⊗µ

  A ⊗ A // A µ

  

η A

⊗A ⊗η

  // oo A ⊗ A A ⊗

  ✶ ⊗ A ✶ ✭✶✳✷✮ l A A µ r $$ zz A. ❖ ♠♦r✜s♠♦ µ é ❝❤❛♠❛❞♦ ❞❡ ▼✉❧t✐♣❧✐❝❛çã♦ ♦✉ Pr♦❞✉t♦ ❡ η é ❝❤❛✲

  ♠❛❞♦ ❞❡ ❯♥✐❞❛❞❡✳ ◗✉❛♥❞♦ µ s❛t✐s❢❛③ ♦ ❞✐❛❣r❛♠❛ ✶✳✶ ❞✐③❡♠♦s q✉❡ µ é ❛ss♦❝✐❛t✐✈♦ ❡ ♦ ❞✐❛❣r❛♠❛ ✶✳✷ é ❝❤❛♠❛❞♦ ❞❡ ❛①✐♦♠❛ ❞❛ ✉♥✐❞❛❞❡✳ ❉❡ ♠❛♥❡✐r❛ ✉s✉❛❧ ✈❛♠♦s ♦♠✐t✐r ♦ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ a A,A,A : (A ⊗ A) ⊗ A −→ A ⊗ (A ⊗ A).

  ❉❡st❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛ ✶✳✶ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ µ

  ⊗A

  // A ⊗ A ⊗ A A ⊗ A . A µ

  ⊗µ

  A ⊗ A // A µ ◆♦ss♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦ é ♥❛ ✈❡r❞❛❞❡ ❛ ♠♦t✐✈❛çã♦ ♣❛r❛ ❛ ❞❡✜♥✐çã♦

  ✶✳✶✻✱ ✐♥❝❧✉s✐✈❡ ♣❛r❛ ♦ ♥♦♠❡ ♠♦♥ó✐❞❡✳ ◆❡st❡ ❡①❡♠♣❧♦✱ ♠♦♥ó✐❞❡ é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ A ♠✉♥✐❞♦ ❞❡ ✉♠❛ ♦♣❡r❛çã♦ ❛ss♦❝✐❛t✐✈❛ q✉❡ ♣♦ss✉✐ A ❡❧❡♠❡♥t♦ ♥❡✉tr♦ e ✳ ❊①❡♠♣❧♦ ✶✳✶✼ ❱❛♠♦s ❝♦♥s✐❞❡r❛r ❛ ❝❛t❡❣♦r✐❛ Set q✉❡ é ♠♦♥♦✐❞❛❧✳ ❖s ♦❜❥❡t♦s á❧❣❡❜r❛s ♥❡ss❛ ❝❛t❡❣♦r✐❛ sã♦ ♠♦♥ó✐❞❡s✳

  ❉❡ ❢❛t♦✱ s❡❥❛ A ✉♠ ♦❜❥❡t♦ á❧❣❡❜r❛ ❡♠ Set✱ ♦✉ s❡❥❛✱ ❡①✐st❡♠ µ : A × A −→ A

  ❡ η : {∗} −→ A✱ q✉❡ s❛t✐s❢❛③❡♠ ✶✳✶ ❡ ✶✳✷✳ ❉❡✜♥✐♥❞♦ µ(a, b) = ab A

  ❡ η(∗) = e ✱ ♦s ❛①✐♦♠❛s ❞❡ ♦❜❥❡t♦ á❧❣❡❜r❛ s❡ tr❛❞✉③❡♠ ❝♦♠♦ (ab)c = a(bc) A a = a = ae A .

  ❡ e P♦rt❛♥t♦✱ A é ✉♠ ♠♦♥ó✐❞❡✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛❞♦ A ✉♠ ♠♦♥ó✐❞❡✱ ♦❜t❡♠♦s ✉♠ ♦❜❥❡t♦ á❧❣❡❜r❛ ❡♠ Set✳ ❊①❡♠♣❧♦ ✶✳✶✽ ◆❛ ❝❛t❡❣♦r✐❛ V ect k ♦s ♦❜❥❡t♦s á❧❣❡❜r❛s sã♦ ❡①❛t❛♠❡♥t❡ ❛s k✲á❧❣❡❜r❛s ✉♥✐t❛✐s✳

  ❉❡ ❢❛t♦✱ s❡ A é ✉♠ ♦❜❥❡t♦ á❧❣❡❜r❛ ❡♠ V ect k ❡①✐st❡♠ ❛♣❧✐❝❛çõ❡s µ : A ⊗ A −→ A, µ(a ⊗ b) = a · b

  ❡ η : k −→ A, η(1 k ) = 1 A k

  ✲❜✐❧✐♥❡❛r❡s✱ ❝♦♠ (a · b) · c = a · (b · c) A · a = a = a · 1 A .

  ❡ 1 P♦rt❛♥t♦✱ A é ✉♠❛ k✲á❧❣❡❜r❛✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ A é ✉♠❛ k✲á❣❡❜r❛✱ ❡♥tã♦ ❞❡✜♥✐♠♦s ✉♠❛ ❛♣❧✐❝❛çã♦ k✲❜✐❧✐♥❡❛r µ : A × A −→ A✱ µ(a, b) = ab ❡ ♣♦rt❛♥t♦✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧✱ t❡♠♦s q✉❡ ❡①✐st❡ ú♥✐❝❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r µ : A ⊗ A −→ A t❛❧ q✉❡ µ(a ⊗ b) = ab.

  ❖ ♠♦r✜s♠♦ ✉♥✐❞❛❞❡ é ❞❛❞♦ ♣♦r η : k −→ A, η(1 k ) = 1 A . ➱ ❢á❝✐❧ ✈❡r✐✜❝❛r ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❞✐❛❣r❛♠❛s ✶✳✶ ❡ ✶✳✷✳ ❊①❡♠♣❧♦ ✶✳✶✾ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ❊♥tã♦ ♦ ♦❜❥❡t♦ ✉♥✐✲ ❞❛❞❡ ✶ é ✉♠ ♦❜❥❡t♦ á❧❣❡❜r❛ ❡♠ C✳

  = r = r ❉❡ ❢❛t♦✱ t❡♠♦s q✉❡ l ✶ ✶ ✱ ✈❡r ❬✶✻❪ ▲❡♠❛✳❳■✳✷✳✸✱ ❞❡✜♥❛ µ := l ✶ ✶

  ❡ η = Id ✶ ✳ ❊♥tã♦ ♣❡❧♦ ❛①✐♦♠❛ ❞♦ tr✐â♥❣✉❧♦✱ ❛♣❧✐❝❛❞♦ ♣❛r❛ ❛ ✉♥✐❞❛❞❡ ♠♦♥♦✐❞❛❧ ✶✱ t❡♠♦s

  µ( ) = µ(r ⊗ ✶ ⊗ µ) = µ(✶ ⊗ l ✶ ✶ ✶) = µ(µ ⊗ ✶),

  ❛ ✉♥✐t❛❧✐❞❛❞❡ é tr✐✈✐❛❧♠❡♥t❡ s❛t✐s❢❡✐t❛✳ A , η A ) B , η B ) ❉❡✜♥✐çã♦ ✶✳✷✵ ❙❡❥❛♠ (A, µ ❡ (B, µ ♦❜❥❡t♦s á❧❣❡❜r❛ ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ C✳ ❯♠ ♠♦r✜s♠♦ ❞❡ ♦❜❥❡t♦s á❧❣❡❜r❛ f : A −→ B

  é ✉♠ ♠♦r✜s♠♦ ❡♠ C✱ t❛❧ q✉❡ ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s ❝♦♠✉t❛♠✳ f ⊗f f //

  A ⊗ A B ⊗ B A // B µ µ η A B A OO >> η B A // B ✶. f k k , ⊗ k , k) M ❙❡❥❛ k ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✉♥✐❞❛❞❡✱ ❡♥tã♦ ❛ ❝❛t❡❣♦r✐❛ ( k k M

  é ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ❊♠ q✉❡ é ❛ ❝❛t❡❣♦r✐❛ ❞♦s k✲❜✐♠ó❞✉❧♦s✳ k k M ❯♠❛ k✲á❧❣❡❜r❛ é ✉♠ ♦❜❥❡t♦ á❧❣❡❜r❛ (R, µ, η) ♥❛ ❝❛t❡❣♦r✐❛ ✳

  R R M P♦❞❡♠♦s ❡♥tã♦ s✉❜✐r ✉♠ ❞❡❣r❛✉ ❡ ❝♦♥s✐❞❡r❛r♠♦s ❛ ❝❛t❡❣♦r✐❛

  ❞♦s R✲❜✐♠ó❞✉❧♦s✱ q✉❡ ❥á ✈✐♠♦s s❡r ♠♦♥♦✐❞❛❧✱ ❝♦♠ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❜❛❧❛♥❝❡❛❞♦ s♦❜r❡ R✱ q✉❡ ♣♦r s✉❛ ✈❡③ é ❛ ✉♥✐❞❛❞❡ ❞❛ ❝❛t❡❣♦r✐❛ ❝✐t❛❞❛✳

  ❆ ♣❛rt✐r ❞❛q✉✐✱ s❡♠♣r❡ q✉❡ ♥ã♦ ❛♣❛r❡❝❡r ✐♥❢♦r♠❛çã♦ s♦❜r❡ R ❡ k✱ ✜❝❛ s✉❜❡♥t❡♥❞✐❞♦ q✉❡ R é ✉♠❛ á❧❣❡❜r❛ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k✳ ❉❡✜♥✐çã♦ ✶✳✷✶ ❯♠ R✲❛♥❡❧ é ✉♠ ♦❜❥❡t♦ á❧❣❡❜r❛ (A, µ, η) ♥❛ ❝❛t❡❣♦r✐❛ R R , ⊗ R , R) M ♠♦♥♦✐❞❛❧ ( ✳ op op op

  , µ , η) P❛r❛ ✉♠ R✲❛♥❡❧ (A, µ, η)✱ ♦ ♦♣♦st♦ s✐❣✐♥✐✜❝❛ ♦ R ✲❛♥❡❧ (A ✳ op op op

  ❊♠ q✉❡ A é ♦ ♠❡s♠♦ k✲♠ó❞✉❧♦✱ A t❡♠ ❡str✉t✉r❛ ❞❡ R ✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ ✭r❡s♣✳ à ❞✐r❡✐t❛✮ ✈✐❛ ❛ R✲❛çã♦ à ❞✐r❡✐t❛ ✭r❡s♣✳ à ❡sq✉❡r❞❛✮✱ op op

  (a ⊗ R

  b) = µ(b ⊗ R

  a) ❛ ♠✉❧t✐♣❧✐❝❛çã♦ é ❞❛❞❛ ♣♦r µ ❡ ❛ ✉♥✐❞❛❞❡ é ❛ ♠❡s♠❛ η✳

  ✶✳✸✳✶ ▼ó❞✉❧♦s s♦❜r❡ ▼♦♥ó✐❞❡s

  ❉❡✜♥✐çã♦ ✶✳✷✷ ❙❡❥❛ (A, µ, η) ✉♠ ♦❜❥❡t♦ á❧❣❡❜r❛ ❡♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ❯♠ ♠ó❞✉❧♦ à ❞✐r❡✐t❛ s♦❜r❡ A✱ ♦✉ ✉♠ A✲♠ó❞✉❧♦ à ❞✐✲ M ) M : M ⊗A → M r❡✐t❛✱ é ✉♠ ♣❛r (M, θ ❡♠ q✉❡ M é ✉♠ ♦❜❥❡t♦ ❡♠ C ❡ θ é ✉♠ ♠♦r✜s♠♦ ❡♠ C✱ t❛❧ q✉❡ ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s ❝♦♠✉t❛♠ M M

  ⊗µ ⊗η

  // // M ⊗ A

  M ⊗ A ⊗ A M ⊗ A M ⊗ θ ⊗A θ M M Mr M θ %% M. M ⊗ A // M, θ M

  ❆♥❛❧♦❣❛♠❡♥t❡✱ ❞❡✜♥✐✲s❡ A✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛✳ ❉❡✜♥✐çã♦ ✶✳✷✸ ❙❡❥❛♠ (A, µ, η) ✉♠ ♦❜❥❡t♦ á❧❣❡❜r❛ ❡♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ M ) N ) ♠♦♥♦✐❞❛❧ ❡ (M, θ ✱ (N, θ ♠ó❞✉❧♦s à ❞✐r❡✐t❛ s♦❜r❡ A✳ ❯♠ ♠♦r✜s♠♦ f : M −→ N

  ❡♠ C é ✉♠ ♠♦r✜s♠♦ ❞❡ A✲♠ó❞✉❧♦s à ❞✐r❡✐t❛ s❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦✿ f ⊗A M ⊗ A // N ⊗ A θ θ M N M // N. f

  ❆♥❛❧♦❣❛♠❡♥t❡✱ ❞❡✜♥✐✲s❡ ♠♦r✜s♠♦ ❞❡ A✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳

  ❊①❡♠♣❧♦ ✶✳✷✹ ❙❡❥❛ (A, µ, η) ✉♠ ♦❜❥❡t♦ á❧❣❡❜r❛ ❡♠ Set✱ ♦✉ s❡❥❛✱ ✉♠ ♠♦♥ó✐❞❡✳ ❊♥tã♦ ❛ ❡str✉t✉r❛ ❞❡ A✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ X é ♦ ♠❡s♠♦ q✉❡ ✉♠❛ ❛çã♦ à ❞✐r❡✐t❛ ❞❡ A ❡♠ X✳ ❯♠❛ ❛çã♦ à ❞✐r❡✐t❛ ❞❡ A ❡♠ X✱ é ✉♠❛ ❛♣❧✐❝❛çã♦ φ : A −→ End(X)✱ q✉❡ s❛t✐s❢❛③ φ(ab) = φ(b) ◦ φ(a)

  ✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ A✱ ❡♠ q✉❡ End(X) é ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❞❡ X ❡♠ X✳ X )

  ❉❡ ❢❛t♦✱ ❝♦♠❡ç❛♠♦s ❝♦♠ ✉♠ A✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ (X, θ ❡ ❞❡✜♥✐♠♦s φ : A −→ End(X) X (x, a)

  ✱ ♣♦r φ(a)(x) = θ ✱ ♣❛r❛ q✉❛✐sq✉❡r x ∈ X ❡

  a, b ∈ A ✳ ❊♥tã♦✱

  φ(ab)(x) = θ X (x, ab) = θ X (x, µ(a, b)) = θ X ◦ (X, µ)(x, a, b) = θ X ◦ (θ X , A)(x, a, b) = θ X (θ X (x, a), b) = φ(b)(φ(a)(x)).

  P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛❞❛ ✉♠❛ ❛çã♦ à ❞✐r❡✐t❛ ❞❡ A ❡♠ X✱ ✈✐❛ φ : A −→ End(X) X (x, a) := φ(a)(x)

  ✱ ❡♥tã♦ θ ❞❡✜♥❡ ✉♠❛ ❡str✉t✉r❛ ❞❡ A✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❡♠ X✳

  ✶✳✹ ❘✲❝♦❛♥é✐s ✭❝♦♠♦♥ó✐❞❡s✮

  ◆❡st❛ s❡çã♦ ✈❛♠♦s ❞❡✜♥✐r ♦ q✉❡ s❡r✐❛ ✉♠❛ ❡①t❡♥sã♦ ❞❛ ♥♦çã♦ ❞❡ ✉♠❛ k ✲❝♦á❧❣❡❜r❛✳ ▼❛✐s ❡①♣❧✐❝✐t❛♠❡♥t❡ ✈❛♠♦s ❞❡✜♥✐r ❛ ♥♦çã♦ ❞❡ R✲❝♦❛♥❡❧✱

  ♦✉ s❡❥❛✱ ✉♠❛ ♥♦çã♦ ❞❡ ❝♦á❧❣❡❜r❛ s♦❜r❡ ✉♠ ❛♥❡❧ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❝♦♠✉t❛t✐✈♦✳ ❉❡✜♥✐çã♦ ✶✳✷✺ ❯♠ ♦❜❥❡t♦ ❝♦á❧❣❡❜r❛ ♦✉ ✉♠ ❝♦♠♦♥ó✐❞❡ ❡♠ ✉♠❛ ❝❛✲ t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ (C, ⊗, ✶, a, l, r) é ✉♠❛ tr✐♣❧❛ (C, ∆, ε)✱ ❡♠ q✉❡ C é ✉♠ ♦❜❥❡t♦ ❡♠ C ❡ ∆ : C −→ C ⊗ C✱ ε : C −→ ✶ sã♦ ♠♦r✜s♠♦s ❡♠ C✱ s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ ❝♦✉♥✐t❛❧✐❞❛❞❡✱ ♦✉ s❡❥❛✱

  ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s ❝♦♠✉t❛♠ l r − − 1 1

  

∆ C C

  oo // C // C ⊗ C C C ⊗ . ✶ ⊗ C ✶ dd ::

  ∆ ∆⊗C ε ⊗C C ⊗ε

  C ⊗ C

  ∆ (C ⊗ C) ⊗ C a C,C,C

  C ⊗ C // C ⊗ (C ⊗ C) C

  ⊗∆

  ❖ ♠♦r✜s♠♦ ∆ é ❝❤❛♠❛❞♦ ❞❡ ❈♦♠✉❧t✐♣❧✐❝❛çã♦ ♦✉ ❈♦♣r♦❞✉t♦ ❡ ε é ❝❤❛♠❛❞♦ ❞❡ ❈♦✉♥✐❞❛❞❡✳ P❛r❛ t♦❞♦ c ∈ C✱ t❡♠♦s ❛ ♥♦t❛çã♦ ❞❡ ❙✇❡❡❞❧❡r ❞❛❞❛ ♣♦r ∆(c) = op c ⊗ c

  

(1) (2) ✳ P❛r❛ ✉♠ R✲❝♦❛♥❡❧ (C, ∆, ε)✱ ♦ s❡✉ ❝♦✲♦♣♦st♦ é ♦ R ✲

cop , ∆ cop , ε) cop cop

  ❝♦❛♥❡❧ (C ✳ ❊♠ q✉❡ C é ♦ ♠❡s♠♦ k✲♠ó❞✉❧♦ C✱ C op t❡♠ ❡str✉t✉r❛ ❞❡ R ✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ ✭r❡s♣✳ à ❞✐r❡✐t❛✮ ✈✐❛ ❛ R✲ ❛çã♦ à ❞✐r❡✐t❛ ✭r❡s♣✳ à ❡sq✉❡r❞❛✮✳ ❆ ❝♦♠✉❧t✐♣❧✐❝❛çã♦ é ❞❛❞❛ ♣♦r op ∆ cop (c) = c ⊗ R c

  (2) (1) ❡ ❛ ❝♦✉♥✐❞❛❞❡ é ❛ ♠❡s♠❛ ε✳

  ❊①❡♠♣❧♦ ✶✳✷✻ ❈♦♥s✐❞❡r❛♥❞♦ ❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❞♦s ❝♦♥❥✉♥t♦s Set✱ t ❡♠♦s q✉❡ t♦❞♦ ❝♦♥❥✉♥t♦ é ✉♠ ♦❜❥❡t♦ ❝♦á❧❣❡❜r❛ ♥❡ss❛ ❝❛t❡❣♦r✐❛✳ ❉❡ ❢❛t♦✱ s❡❥❛ X ✉♠ ❝♦♥❥✉♥t♦ q✉❛❧q✉❡r✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧

  ❞♦ ♣r♦❞✉t♦ ❡♠ Set✱ t❡♠♦s q✉❡ ❡①✐st❡ ú♥✐❝♦ ∆ : X −→ X × X✱ t❛❧ q✉❡✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦ Id Id X X , X

  ∆

  {{ ## oo // X

  X X × X π π 1 2 ♦✉ s❡❥❛✱ ∆(x) = (x, x) é ❛ ❛♣❧✐❝❛çã♦ ❞✐❛❣♦♥❛❧✳ ❚❛♠❜é♠✱ t❡♠♦s q✉❡ ♣❛r❛ t♦❞♦ X ∈ Set ❡①✐st❡ ✉♠ ú♥✐❝♦ ε : X −→ {∗}✱ ♣♦✐s {∗} é ♦❜❥❡t♦ ✜♥❛❧ ❡♠ Set✳ ❊①❡♠♣❧♦ ✶✳✷✼ ❈♦♥s✐❞❡r❛♥❞♦ ❛ ❝❛t❡❣♦r✐❛ V ect K ❞♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s s♦❜r❡ ♦ ❝♦r♣♦ k✳ ❚❡♠♦s q✉❡ ✉♠ ♦❜❥❡t♦ ❝♦á❧❣❡❜r❛ ❡♠ V ect K é ✉♠❛ k

  ✲❝♦á❧❣❡❜r❛ ♥♦ s❡♥t✐❞♦ tr❛❞✐❝✐♦♥❛❧✳ ❊①❡♠♣❧♦ ✶✳✷✽ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ❊♥tã♦ ❛ ✉♥✐❞❛❞❡ ♠♦✲ ♥♦✐❞❛❧ ✶ é ✉♠ ♦❜❥❡t♦ ❝♦á❧❣❡❜r❛ ❡♠ C✳

  

−1 −1

  = r ❉❡ ❢❛t♦✱ ❜❛st❛ ❞❡✜♥✐r ∆ = l ❡ ε = Id ✳ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r✱

  ✶

✶ ✶

  ✉s❛♥❞♦ ▲❡♠❛✳❳■✳✷✳✸ ❡♠ ❬✶✻❪✱ q✉❡ ❛ ❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ ❛ ❝♦✉♥✐t❛❧✐❞❛❞❡ sã♦ s❛t✐s❢❡✐t❛s✳ C , ε C ) D , ε D ) ❉❡✜♥✐çã♦ ✶✳✷✾ ❙❡❥❛♠ (C, ∆ ❡ (D, ∆ ♦❜❥❡t♦s ❝♦á❧❣❡❜r❛s ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ C✳ ❯♠ ♠♦r✜s♠♦ ❞❡ ♦❜❥❡t♦s ❝♦á❧❣❡❜r❛s f : C −→ D

  é ✉♠ ♠♦r✜s♠♦ ❡♠ C t❛❧ q✉❡ ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s sã♦ ❝♦♠✉t❛t✐✈♦s✿ f f

  // // C D C D ε

  ∆ ∆ C C D ε D

  ~~ C ⊗ C // D ⊗ D f ✶.

  ⊗f

  ❉❡✜♥✐çã♦ ✶✳✸✵ ❯♠ R✲❝♦❛♥❡❧ é ✉♠ ♦❜❥❡t♦ ❝♦á❧❣❡❜r❛ (C, ∆, ε) ♥❛ ❝❛✲ R R , ⊗ R , R) M t❡❣♦r✐❛ ( ❞♦s R✲❜✐♠ó❞✉❧♦s✳ ❊①❡♠♣❧♦ ✶✳✸✶ ❖ ❡①❡♠♣❧♦ tr✐✈✐❛❧ ❞❡ R✲❝♦❛♥❡❧ é ❥✉st❛♠❡♥t❡ ❛ k✲á❧❣❡❜r❛ s✉❜❥❛❝❡♥t❡ R✳

  −1 −1

  = r : R −→ R ⊗ R R ❉❡ ❢❛t♦✱ ❜❛st❛ ❞❡✜♥✐r♠♦s ∆ = l ❡ ε = R R

  Id R : R −→ R ✳ ➱ tr✐✈✐❛❧ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❞✐❛❣r❛♠❛s✳

  ❊①❡♠♣❧♦ ✶✳✸✷ ❙❡❥❛ G ✉♠ ❣r✉♣♦✱ R ✉♠❛ k✲á❧❣❡❜r❛ G✲❣r❛❞✉❛❞❛✱ ♦✉ = ⊕ g R g g ⊗ R h ) ⊆ R gh s❡❥❛✱ R ∼ ∈G ❡ µ(R ✳ ❱❛♠♦s ❝♦♥str✉✐r ✉♠ R✲❝♦❛♥❡❧

  C := ⊕ g R δ

  ∈G g ✳

  ❉❡ ❢❛t♦✱ ❞❛♠♦s ❛ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦s ♣❛r❛ C ♣♦r

  X

  

′ ′ ′

  r · δ g · r = rr δ gh , ∀ r, r ∈ R, h h g R g ∈G ❡st❛ s♦♠❛ é ✜♥✐t❛ ♣♦✐s r ∈ ⊕ ∈G ✱ ❧♦❣♦ ❛♣❡♥❛s ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ❝♦♠♣♦♥❡♥t❡s é ♥ã♦ ♥✉❧❛✳ ❈❧❛r♦ q✉❡ C ❥á t❡♠ ❡str✉t✉r❛ ❞❡ R✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛✱ ❜❛st❛ ✈❡r✐✜❝❛r♠♦s q✉❡ t❡♠ ❡str✉t✉r❛ à ❞✐r❡✐t❛✳ ❉❡ ❢❛t♦✱

  ′

  ∈ R ♣❛r❛ q✉❛✐q✉❡r r, r ❡ g, h ∈ G t❡♠♦s

  X

  ′ ′

  (δ g · r) · r = ( r h δ gh ) · r h

  ∈G

  X X

  ′

  = r h r δ ghk h k k

  ∈G ∈G

  X X

  ′ −1 1

  = r h r δ gl , l l h h l ✉s❛♥❞♦ hk = l, k = h

  ∈G ∈G

  X

  ′

  = (rr ) l δ gl l

  ∈G ′

  = δ g · (rr ). R C g ) = δ g ⊗ δ g ❉❡✜♥✐♠♦s ❡♥tã♦ ∆ : C −→ C ⊗ ✱ ∆(δ ❡ ε : C −→ R✱

  ε(δ g ) = 1 R ✳ ❱❛♠♦s ✈❡r q✉❡ ∆ é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✱ ❝❧❛r♦ q✉❡✱ só

  ♣r❡❝✐s❛♠♦s ♠♦str❛r à ❞✐r❡✐t❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R ❡ g, h ∈ G✱ t❡♠♦s

  X ∆(δ g · r) = ∆( r h δ gh ) h

  ∈G

  X = r h δ gh ⊗ δ gh , h

  

∈G

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ ∆(δ g ) · r = (δ g ⊗ δ g ) · r

  X = δ g ⊗ r h δ gh h

  ∈G

  X = δ g · r h ⊗ δ gh h

  ∈G

  X X = (r h ) k δ gk ⊗ δ gh h k

  

∈G ∈G

  X = δ h,k r h δ gk ⊗ δ gh h,k

  

∈G

  X = r h δ gh ⊗ δ gh , h

  ∈G h ) k = δ h,k r h h ∩ R k = {0}

  ♥❛ ♣❡♥ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ✉s❛♠♦s q✉❡ (r ♣♦✐s R s❡ h 6= k✳ ❊①❡♠♣❧♦ ✶✳✸✸ ✭R✲❝♦❛♥❡❧ ❞❡ ❝♦❛çã♦✮ ❙❡❥❛♠ k ✉♠ ❝♦r♣♦✱ H ✉♠❛ k✲ ❜✐á❧❣❡❜r❛ ❡ R ✉♠ H✲❝♦♠ó❞✉❧♦ á❧❣❡❜r❛ à ❞✐r❡✐t❛✳ ❖✉ s❡❥❛✱ ♣❛r❛ q✉❛✐sq✉❡r

  a, b ∈ R k H ✱ ✉s❛♥❞♦ ♥♦t❛çã♦ ❞❛ ❝♦❛çã♦ ρ : R −→ R ⊗ ✱ ♣♦r ρ(a) =

  (0) (1)

  a ⊗ a ✱ t❡♠♦s

  (0) (1)

  • a ε(a ) = a

  ❀

  (0)(0) (0)(1) (1) (0) (1) (1)

  • a ⊗ a ⊗ a = a ⊗ a ⊗ a

  (1) (2) ❀ (0) (1) (0) (0) (1) (1)

  • (ab) ⊗ (ab) = a b ⊗ a b

  ❀

  • 1

  (2)

  ) = (a ⊗ h

  (1)

  b

  (2)

  · (1 R ⊗ h

  (0)

  ) ⊗ R b

  (1)

  ) = (a ⊗ h

  (1)

  b

  ⊗ R (1 R ⊗ h

  ) ⊗ R (b

  (0)

  ) · b

  (1)

  ) = (a ⊗ h

  (1)

  b

  (2)

  ) ⊗ R (1 R ⊗ h

  (0)(1)

  b

  (1)

  (1)

  (0)

  (0)(0)

  ) = ab

  (1)

  ε(b

  (0)

  ) = ab

  (1)

  ε(h)ε(b

  (0)

  ) = ab

  (1)

  ε(hb

  (0)

  (1)

  ⊗ h

  ⊗ hb

  (0)

  ❡ ε((a ⊗ h) · b) = ε(ab

  ) · b = ∆(a ⊗ h) · b

  (2)

  ) ⊗ R (1 R ⊗ h

  (1)

  ) = (a ⊗ h

  (1)

  b

  (2)

  ⊗ h

  ) = (ab

  (0) R

  (1)

  ⊗ h(bc)

  (0)

  ) = a(bc)

  (1)

  c

  (1)

  ⊗ hb

  (0)

  c

  (0)

  ) · c = (ab

  ⊗ hb

  ❉❡✜♥✐♠♦s ❛❣♦r❛ ∆ : C −→ C ⊗ R C ♣♦r ∆(a ⊗ h) = (a ⊗ h (1) ) ⊗ R (1 R ⊗ h

  (0)

  ((a ⊗ h) · b) · c = (ab

  ✱ ♣❛r❛ q✉❛✐sq✉❡r a, b, c ∈ R ❡ h ∈ H ✳ ❉❡ ❢❛t♦✱ ♠♦str❡♠♦s ❛♣❡♥❛s à ❞✐r❡✐t❛✱ ♣♦✐s à ❡sq✉❡r❞❛ é ❛♥á❧♦❣♦✱

  (1)

  ⊗ hc

  (0)

  ❞❛❞❛ ♣♦r a · (b ⊗ h) · c = abc

  ❊♥tã♦ C = R ⊗ k H ✱ t❡♠ ❡str✉t✉r❛ ❞❡ R✲❝♦❛♥❡❧✳ ❉❡ ❢❛t♦✱ ♣r✐♠❡✐r♦ ♥♦t❡ q✉❡ R ⊗ k H t❡♠ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦

  = 1 R ⊗ 1 H ✳

  (1) R

  ⊗ 1

  (1) = (a ⊗ h) · (bc).

  (2)

  (1) (2)

  (1)

  b

  (2)

  ) ⊗ R (1 R ⊗ h

  (1) (1)

  b

  

(1)

  ⊗ h

  (0)

  ) = (ab

  (2)

  )

  ) ⊗ R (1 R ⊗ (hb

  ) ❡ ε : C −→ R ♣♦r ε(a ⊗ h) = aε(h)✳ ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ∆

  (1)

  )

  

(1)

  ⊗ (hb

  (0)

  ) = (ab

  (1)

  ⊗ hb

  (0)

  ∆((a ⊗ h) · b) = ∆(ab

  ❡ ε sã♦ ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ R ❡ h ∈ H ✱ t❡♠♦s

  )ε(h) = abε(h) = aε(h)b = ε(a ⊗ h)b.

  ◗✉❡ ∆ ❡ ε sã♦ ♠♦r✜s♠♦s à ❡sq✉❡r❞❛ é tr✐✈✐❛❧✳ ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ❛ ❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ ❛ ❝♦✉♥✐t❛❧✐❞❛❞❡ sã♦ s❛t✐s❢❡✐t❛s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a ∈ R ❡ h ∈ H✱ t❡♠♦s

  (∆ ⊗ R C)∆(a ⊗ h) = (∆ ⊗ R C)((a ⊗ h ) ⊗ R (1 R ⊗ h )

  (1) (2)

  = ∆(a ⊗ h ) ⊗ R (1 R ⊗ h )

  (1) (2)

  = (a ⊗ h ) ⊗ R (1 R ⊗ h ) ⊗ R (1 R ⊗ h )

  (1)(1) (1)(2) (2)

  = (a ⊗ h ) ⊗ R (1 R ⊗ h ) ⊗ R (1 R ⊗ h )

  (1) (2)(1) (2)(2)

  = (a ⊗ h ) ⊗ R ∆(1 R ⊗ h )

  (1) (2)

  = (C ⊗ R ∆)∆(a ⊗ h), t❡♠♦s t❛♠❜é♠ (ε ⊗ R C)∆(a ⊗ h) = ε(a ⊗ h ) · (1 R ⊗ h )

  (1) (2)

  = (aε(h )) · (1 R ⊗ h )

  (1) (2)

  = aε(h ) ⊗ h

  (1) (2)

  = a ⊗ ε(h )h

  (1) (2) R ε)∆(a ⊗ h) = a ⊗ h = a ⊗ h, q✉❡ (C ⊗ é ❛♥á❧♦❣♦✳

  ❆♥t❡s ❞♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦ ✈❛♠♦s ❞❡✜♥✐r ✉♠ ♠ó❞✉❧♦ ♣r♦❥❡t✐✈♦ ✜✲ ♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱ q✉❡ s❡rá ✐♠♣♦rt❛♥t❡ ❡♠ ✈ár✐♦s r❡s✉❧t❛❞♦s✱ ♣r✐♥❝✐♣❛❧✲ ♠❡♥t❡ q✉❛♥❞♦ tr❛t❛r♠♦s ❞❛ ❞✉❛❧✐❞❛❞❡✳ ❙❡❥❛ P ✉♠ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✱

  ∗

  = Hom R (P, R) ❝♦♥s✐❞❡r❡♠♦s P ✳

  ∗

  ◆♦t❡ q✉❡ P t❡♠ ❡str✉t✉r❛ ❞❡ R✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛

  ∗

  q✉❛✐sq✉❡r ϕ ∈ P ✱ s, r ∈ R ❡ p ∈ P ✱ ❞❡✜♥✐♠♦s (r · ϕ)(p) = rϕ(p)✱ ❞❡st❛ ❢♦r♠❛ t❡♠♦s (s · (r · ϕ))(p) = s(r · ϕ)(p) = srϕ(p) = ((sr) · ϕ)(p).

  ◆♦t❡ q✉❡ r · ϕ ❝♦♥t✐♥✉❛ R✲❧✐♥❡❛r à ❞✐r❡✐t❛✱ ♦✉ s❡❥❛✱ (r · ϕ)(p · s) = rϕ(p · s) = rϕ(p)s = (r · ϕ)(p)s.

  ∗

  = ❉❡✜♥✐çã♦ ✶✳✸✹ ❙❡❥❛ P ✉♠ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❈♦♥s✐❞❡r❛♥❞♦ P Hom R (P, R)

  ✱ ❞✐③❡♠♦s q✉❡ P é ♣r♦❥❡t✐✈♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ s❡ ❡①✐s✲ n i n i ∈ P } ∈ P } ∗ t❡♠ n ∈ N✱ {x i ❡ {f i ✱ t❛✐s q✉❡✱ ♣❛r❛ t♦❞♦ p ∈ P n =1 =1

  P i n x i · f (p). i ∈ P } t❡♠♦s p = ◆❡st❡ ❝❛s♦ ❞✐③❡♠♦s q✉❡ {x i ❡ i

  =1 i n =1 ∗

  {f ∈ P } i é ❛ ❜❛s❡ ❞✉❛❧ ❞❡ P ✳

  =1

  P n i ϕ(x i ) · f

  ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ ϕ = ♣❛r❛ i =1

  ∗

  t♦❞♦ ϕ ∈ P ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ p ∈ P t❡♠♦s n n

  X i i

  X ( ϕ(x i ) · f )(p) = (ϕ(x i ) · f )(p) i i

  =1 =1 n

  X i = ϕ(x i )f (p) i

  =1 n

  X i = ϕ( x i · f (p)) i =1 = ϕ(p).

  ❊①❡♠♣❧♦ ✶✳✸✺ ✭❈♦❛♥❡❧ ❞❡ ✉♠ R✲♠ó❞✉❧♦ ♣r♦❥❡t✐✈♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✮ B R M ❙❡❥❛♠ B k✲á❧❣❡❜r❛ ❡ P ∈ ♣r♦❥❡t✐✈♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ❝♦♠♦ R✲

  ∗ ∗

  = Hom R (P, R) ∈ ♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❈♦♥s✐❞❡r❡♠♦s P ✳ ◆♦t❡ q✉❡ P R B M

  ✱ ❝♦♠ ❡str✉t✉r❛ ❞❛❞❛ ♣♦r (r · ϕ ↼ b)(p) = rϕ(b · p)✳ ❊♥tã♦✱

  ∗

  C = P ⊗ B P é ✉♠ R✲❝♦❛♥❡❧✳

  ∗

  ⊗ B P ❉❡ ❢❛t♦✱ ♥♦t❡ q✉❡ P é ✉♠ R✲❜✐♠ó❞✉❧♦✱ ❛ ❡str✉t✉r❛ é ❞❛❞❛

  ′ ′ ′ ∗

  = r · ϕ ⊗ p · r ∈ R ♣♦r r · (ϕ ⊗ p) · r ♣❛r❛ q✉❛✐sq✉❡r r, r ✱ ϕ ∈ P ❡ p ∈ P R C

  ✳ ❉❡✜♥✐♠♦s ∆ : C −→ C ⊗ ♣♦r n

  X i ∆(ϕ ⊗ p) = (ϕ ⊗ x i ) ⊗ R (f ⊗ p), i i n i n =1

  } } ❡♠ q✉❡ {x i =1 ❡ {f i =1 é ❛ ❜❛s❡ ❞✉❛❧ ❞❡ P ✱ ❡ ❞❡✜♥✐♠♦s ε : C −→ R

  ∗

  ♣♦r ε(ϕ ⊗ p) = ϕ(p)✱ ♣❛r❛ q✉❛✐sq✉❡r p ∈ P ❡ ϕ ∈ P ✳ ❱❛♠♦s ✈❡r q✉❡

  ∗

  ∆ ❡ ε sã♦ ♠♦r✜s♠♦s ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r ϕ ∈ P ✱ p ∈ P

  ❡ r ∈ R✱ ♠♦str❡♠♦s ❛♣❡♥❛s à ❞✐r❡✐t❛✱ à ❡sq✉❡r❞❛ é ❛♥á❧♦❣♦✳ ❙❡❣✉❡ ❡♥tã♦

  ∆((ϕ ⊗ p) · r) = ∆(ϕ ⊗ p · r) n

  X i = (ϕ ⊗ x i ) ⊗ R (f ⊗ p · r) i

  =1 n

  X i = (ϕ ⊗ x i ) ⊗ R (f ⊗ p) · r i

  =1 n

  X i = ( (ϕ ⊗ x i ) ⊗ R (f ⊗ p)) · r i

  =1

  = ∆(ϕ ⊗ p) · r

  ❡ ε((ϕ ⊗ p) · r) = ε(ϕ ⊗ p · r)

  =1

  ∆(ϕ ⊗ x j ) ⊗ R (f j ⊗ p) = n

  X i =1 n

  X j =1 (ϕ ⊗ x i ) ⊗ R (f i ⊗ x j ) ⊗ R (f j ⊗ p).

  ❚❛♠❜é♠ t❡♠♦s q✉❡ (ε ⊗ R C)∆(ϕ ⊗ B p) = (ε ⊗ R

  C) n

  X j

  (ϕ ⊗ x j ) ⊗ R (f j ⊗ p) = n

  X j

  X i

  =1

  ε(ϕ ⊗ x i ) · (f i ⊗ p) = n

  X i =1 (ε(ϕ ⊗ x i ) · f i ⊗ p)

  = n

  X i

  =1

  =1

  = n

  = ϕ(p · r) = ϕ(p)r = ε(ϕ ⊗ p)r.

  X i

  ▼♦str❡♠♦s ❛❣♦r❛ ❛ ❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞❡ ∆ ❡ ❛ ❝♦✉♥✐t❛❧✐❞❛❞❡ ❞❡ ε✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r ϕ ∈ P

  ∗

  ❡ p ∈ P ✱ t❡♠♦s (C ⊗ R ∆)∆(ϕ ⊗ B p) = (C ⊗ R ∆)( n

  X i

  =1

  (ϕ ⊗ x i ) ⊗ R (f i ⊗ p) = n

  =1

  X j =1 (ϕ ⊗ x j ) ⊗ R (f j ⊗ p))

  (ϕ ⊗ x i ) ⊗ R ∆(f i ⊗ p) = n

  X i

  =1 n

  X j

  =1

  (ϕ ⊗ x i ) ⊗ R (f i ⊗ x j ) ⊗ R (f j ⊗ p), ♣♦r ♦✉tr♦ ❧❛❞♦✱

  (∆ ⊗ R C)∆(ϕ ⊗ B p) = (∆ ⊗ R C)( n

  (ϕ(x i ) · f i ⊗ p) n

  X i = ( (ϕ(x i ) · f ) ⊗ p i

  =1

  = ϕ ⊗ p, ❡ n

  X j (C ⊗ ε)∆(ϕ ⊗ p) = (C ⊗ ε)( (ϕ ⊗ x j ) ⊗ R (f ⊗ p)) j n =1

  X j = (ϕ ⊗ x j ) · ε(f ⊗ p) j

  =1 n

  X i = (ϕ ⊗ x i · f (p)) i

  =1

n

  X i = ϕ ⊗ ( x i · f (p)) i =1 = ϕ ⊗ p.

  ✶✳✹✳✶ ❈♦♠ó❞✉❧♦s ❙♦❜r❡ ❈♦♠♦♥ó✐❞❡s

  ❉❡✜♥✐çã♦ ✶✳✸✻ ❙❡❥❛ C = (C, ∆, ε) ✉♠ ♦❜❥❡t♦ ❝♦á❧❣❡❜r❛ ❡♠ ✉♠❛ ❝❛t❡✲ M )

  ❣♦r✐❛ ♠♦♥♦✐❞❛❧ C✳ ❯♠ C✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛ é ✉♠ ♣❛r (M, ρ ✱ ❡♠ M : M −→ M ⊗ C q✉❡ M é ✉♠ ♦❜❥❡t♦ ❡♠ C ❡ ρ é ✉♠ ♠♦r✜s♠♦ ❡♠ C✱ t❛❧ q✉❡ ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s ❝♦♠✉t❛♠ ρ ρ M M ρ M ⊗∆ M ⊗ε M M // M ⊗ C M // M ⊗ C r M 1

  ## M ⊗ C // M ⊗ C ⊗ C M ⊗ ρ M ✶.

  ⊗C

  ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❞❡✜♥❡✲s❡ C✲❝♦♠ó❞✉❧♦ à ❡sq✉❡r❞❛✳ ❱❛♠♦s ✉s❛r ❛ ♥♦t❛çã♦ ❞❡ ❙✇❡❡❞❧❡r M

  (0) (1)

  ρ (m) = m ⊗ m ∈ M ⊗ C, ( ❡str✉t✉r❛ à ❞✐r❡✐t❛)

  ❡ M (−1) (0) λ (m) = m ⊗ m ∈ C ⊗ M, ( ❡str✉t✉r❛ à ❡sq✉❡r❞❛). ❖s ❛①✐♦♠❛s ❞❡ C✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛ ♥❛ ♥♦t❛çã♦ ❛❝✐♠❛ ✜❝❛♠✿

  (0)(0) (0)(1) (1) (0) (1) (1)

  • m ⊗ m ⊗ m = m ⊗ m ⊗ m

  (1) (2) ❀ (0) (1)

  • m · ε(m ) = m

  ✳ ❉❡✜♥✐çã♦ ✶✳✸✼ ❙❡❥❛♠ C ✉♠ R✲❝♦❛♥❡❧✱ M ❡ N C✲❝♦♠ó❞✉❧♦s à ❞✐✲ r❡✐t❛✳ ❯♠ ♠♦r✜s♠♦ ❞❡ C✲❝♦♠ó❞✉❧♦s à ❞✐r❡✐t❛ é ✉♠ ♠♦r✜s♠♦ ❞❡ R

  ✲❜✐♠ó❞✉❧♦s f : M −→ N t❛❧ q✉❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦✿ f // ρ ρ M N M N

  M ⊗ R C // N ⊗ R f C C.

  

R

  ❊①❡♠♣❧♦ ✶✳✸✽ ❙❡❥❛ C ✉♠ R✲❝♦❛♥❡❧✳ ❊♥tã♦ C t❡♠ ✉♠❛ ❡str✉t✉r❛ tr✐✲ ✈✐❛❧ ❞❡ C✲❝♦♠ó❞✉❧♦ ✭à ❞✐r❡✐t❛ ♦✉ à ❡sq✉❡r❞❛✮ ❡♠ q✉❡ ❛ ❝♦❛çã♦ é ✐❣✉❛❧ ❛♦ ❝♦♣r♦❞✉t♦✳ ❉❡✜♥✐çã♦ ✶✳✸✾ ❙❡❥❛ (C, ∆, ε) ✉♠ R✲❝♦❛♥❡❧✳ ❯♠ ❡❧❡♠❡♥t♦ g ∈ C é R ❞✐t♦ s❡r ✉♠ ❣r♦✉♣✲❧✐❦❡ s❡ ∆(g) = g ⊗ g ❡ ε(g) = 1 ✳ ❊①❡♠♣❧♦ ✶✳✹✵ ❙❡❥❛ C ✉♠ R✲❝♦❛♥❡❧✳ ❙❡ ❡①✐st❡ g ∈ C ❣r♦✉♣✲❧✐❦❡✱ ❡♥tã♦ R R

  : R −→ R ⊗ C ∼ é ✉♠ C✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❉❡✜♥✐♠♦s ❛ ❝♦❛çã♦ ρ = R

  C (r) = r ⊗ g ∼ ✱ ♣♦r ρ = r · g ♣❛r❛ t♦❞♦ r ∈ R✳ ❖✉ s✐♠❡tr✐❝❛♠❡♥t❡ R é R

  : R −→ C ⊗R ∼ ✉♠ C✲❝♦♠ó❞✉❧♦ à ❡sq✉❡r❞❛✳ ❉❡✜♥✐♠♦s ❛ ❝♦❛çã♦ ρ = C ✱ R

  (r) = g ⊗ r ∼ ♣♦r ρ = g · r ♣❛r❛ t♦❞♦ r ∈ R✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ R é ✉♠ C

  ✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛✱ ❡♥tã♦ C ♣♦ss✉✐ ✉♠ ❣r♦✉♣✲❧✐❦❡ g✱ ❝♦♥❢♦r♠❡ ▲❡♠❛ ✶✳✺✶✳ ❊①❡♠♣❧♦ ✶✳✹✶ ❙❡❥❛♠ G ✉♠ ❣r✉♣♦ ❡ M ✉♠ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ G✲ g M g g ❣r❛❞✉❛❞♦✱ ♦✉ s❡❥❛✱ M ∼ = ⊕ ∈G ✱ ❡♠ q✉❡ M sã♦ R✲♠ó❞✉❧♦s à ❞✐r❡✐t❛✳ g Rδ g ❈♦♥s✐❞❡r❡♠♦s RG = ⊕ ∈G ✱ ♦ ❝♦❛♥❡❧ ❞❡ ❣r✉♣♦ ✭✈❡r ❊①❡♠♣❧♦ ✶✳✸✷✮✱

  ′

  ∈ R ❝♦♠ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦s ❞❛❞❛✱ ♣❛r❛ q✉❛✐sq✉❡r r, r ❡ g ∈ G✱ g · r = rr δ g ′ ′

  ♣♦r r · δ ✱ ❡ ❡str✉t✉r❛ ❞❡ R✲❝♦❛♥❡❧ ❞❛❞❛ ♣♦r ∆ : RG −→ RG ⊗ R RG g ) = δ g ⊗ δ g g ) = 1 R

  ✱ ∆(δ ❡ ε : RG −→ R✱ ε(δ ✳ ❊♥tã♦✱ M é R RG ✉♠ RG✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❝♦♠ ❝♦❛çã♦ ρ : M −→ M ⊗ ❞❡✜♥✐❞❛

  P P m g ) = m g ⊗ δ g ♣♦r ρ( g g ✳

  ∈G ∈G ✶✳✺ ❉✉❛❧✐❞❛❞❡

  ❙❛❜❡♠♦s q✉❡ ♦ ❞✉❛❧ ❞❡ ✉♠❛ k✲❝♦á❧❣❡❜r❛✱ t❡♠ ✉♠❛ ❡str✉t✉r❛ ❝❛♥ô✲ ♥✐❝❛ ❞❡ k✲á❧❣❡❜r❛✳ ❉❡ ♠❛♥❡✐r❛ ✐♥✈❡rs❛✱ s❡ ✉♠❛ k✲á❧❣❡❜r❛ é ♣r♦❥❡t✐✈❛

  ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞❛ ❝♦♠♦ k✲♠ó❞✉❧♦✱ ❡♥tã♦ ❡ss❛ k✲á❧❣❡❜r❛ ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ❞❡ k✲❝♦á❧❣❡❜r❛ ✈❡r ❬✶✶❪✳ ■r❡♠♦s ❢❛③❡r ♦ ❛♥á❧♦❣♦ ♣❛r❛ R✲❛♥é✐s ❡ R✲❝♦❛♥é✐s✳ Pr♦♣♦s✐çã♦ ✶✳✹✷ P❛r❛ ✉♠ R✲❝♦❛♥❡❧ (C, ∆, ε)✱ ♦ s❡✉ ❞✉❛❧ à ❡sq✉❡r❞❛

  ∗

  C := R Hom(C, R) ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ❝❛♥ô♥✐❝❛ ❞❡ R✲❛♥❡❧✱ ❡♠ q✉❡

  ∗

  C ♣❛r❛ q✉❛✐sq✉❡r ϕ, ψ ∈ ✱ c ∈ C ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ✭♣r♦❞✉t♦ ❞❡ ❝♦♥✈♦❧✉✲

  · ϕ(c )) C = ε çã♦✮ é ❞❛❞❛ ♣♦r (ϕ ∗ ψ)(c) := ψ(c (1)

  2 ❡ ✉♥✐❞❛❞❡ ✶ ✳ ∗

  C ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ ❛ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦ ❡♠ é

  ′ ′ ′ ∗

  )(c) = ϕ(c · r)r ∈ R C ❞❛❞❛ ♣♦r (r ⇀ ϕ · r ♣❛r❛ q✉❛✐sq✉❡r r, r ✱ ϕ ∈ ❡ c ∈ C✳ ❆❣♦r❛ ✈❛♠♦s ♠♦str❛r q✉❡ ♦ ♣r♦❞✉t♦ ❞❡ ❝♦♥✈♦❧✉çã♦✱ ❞❡✜♥✐❞♦

  ∗

  C ❛❝✐♠❛✱ é R✲❜❛❧❛♥❝❡❛❞♦✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r ϕ, ψ ∈ ✱ c ∈ C ❡ r ∈ R

  ✱ t❡♠♦s ((ϕ · r) ∗ ψ)(c) = ψ(c · (ϕ · r)(c ))

  (1) (2)

  = ψ(c · (ϕ(c )r))

  (1) (2)

  = ψ((c · ϕ(c )) · r)

  (1) (2)

  = (r ⇀ ψ)(c · ϕ(c ))

  (1) (2) = (ϕ ∗ (r ⇀ ψ))(c).

  ❆❣♦r❛✱ é ❝❧❛r♦ q✉❡ ♦ ♣r♦❞✉t♦ ❞❡ ❝♦♥✈♦❧✉çã♦ é ♠♦r✜s♠♦ ❞❡ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✱ ❜❛st❛ ♠♦str❛r♠♦s ♣❛r❛ ❛ ❡str✉t✉r❛ à ❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱ ❛♥t❡s ♥♦t❡ q✉❡ ∆ é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✱ ❛ss✐♠ t❡♠♦s✱ ∆(c · r) = ∆(c) · r = (c ⊗ c ) · r = c ⊗ (c · r).

  (1) (2) (1) (2)

  ❉❡st❛ ❢♦r♠❛✱ ((r ⇀ ϕ) ∗ ψ)(c) = ψ(c · (r ⇀ ϕ(c )))

  (1) (2)

  = ψ(c · ϕ(c · r)),

  (1) (2)

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ (r ⇀ (ϕ ∗ ψ))(c) = (ϕ ∗ ψ)(c · r) = ψ(c · ϕ(c · r)). C = ε (1) (2)

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ✶ ❡ q✉❡ ♦ ♣r♦❞✉t♦ ❞❡ ❝♦♥✈♦❧✉çã♦ é ❛ss♦❝✐✲

  ∗

  C ❛t✐✈♦✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r c ∈ C✱ ϕ, ψ ❡ θ ∈ ✱ t❡♠♦s

  (ε ∗ ϕ)(c) = ϕ(c · ε(c ))

  (1) (2)

  = ϕ(c)

  ❡ (ϕ ∗ ε)(c) = ε(c · ϕ(c ))

  (1) (2)

  = ε(c )ϕ(c )

  (1) (2)

  = ϕ(ε(c ) · c )

  (1) (2) = ϕ(c).

  · ϕ(c )) = c ⊗ (c · ϕ(c )) ❆❣♦r❛ ♥♦t❡ q✉❡✱ ∆(c (1) (2) (1) (2) (3) ✱ ❞❡st❛ ❢♦r♠❛✱ t❡♠♦s

  (ϕ ∗ (ψ ∗ θ))(c) = (ψ ∗ θ)(c · ϕ(c ))

  (1) (2)

  = θ(c · (ψ(c · ϕ(c ))))

  (1) (2) (3)

  = θ(c · (ψ ∗ ϕ)(c ))

  (1) (2) = ((ϕ ∗ ψ) ∗ θ)(c).

  ❆❣♦r❛ ✈❡r❡♠♦s ❞♦✐s ❧❡♠❛s q✉❡ s❡rã♦ ✉s❛❞♦s ♥❛ ♥♦ss❛ ♣ró①✐♠❛ ♣r♦✲ ♣♦s✐çã♦✳ ▲❡♠❛ ✶✳✹✸ ❙❡❥❛ B ✉♠ R✲❛♥❡❧ ♣r♦❥❡t✐✈♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ❝♦♠♦ R✲

  

  = Hom R (B, R) ♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❈♦♥s✐❞❡r❛♥❞♦ B t❡♠♦s q✉❡ ❛ s❡✲ ❣✉✐♥t❡ ❛♣❧✐❝❛çã♦ é ✉♠❛ ❜✐❥❡çã♦✿

  ∗ ∗

  d ( ) : B ⊗ R B → Hom R (B ⊗ R

  B, R) \

  ϕ ⊗ ψ 7→ ϕ ⊗ ψ ϕ ⊗ ψ(a ⊗ b) = ϕ(ψ(a) · b)

  ❡♠ q✉❡ \ ✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ B✳ ❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ ❞❡✜♥❛

  ∗ ∗

  g R R R ( ) : Hom (B ⊗

  B, R) −→ B ⊗ B , F 7−→ e F t❛❧ q✉❡ n

  X e F := F (x i ⊗ x j ) · ρ j ⊗ ρ i , i,j n n =1 i ∈ B} i ∈ B }

  ❡♠ q✉❡ {x i ✱ {ρ i é ❛ ❜❛s❡ ❞✉❛❧✳ ▼♦str❡♠♦s q✉❡

  =1 =1

  d ( ) ( )

  ❡ g sã♦ ✐♥✈❡rs❛s ✉♠❛ ❞❛ ♦✉tr❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r F ∈ Hom R (B ⊗ R

  B, R) ❡ a, b ∈ B✱ t❡♠♦s

   \  n

  X be F (a ⊗ b) =  F (x i ⊗ x j ) · ρ j ⊗ ρ i (a ⊗ b) i,j

  =1

  = n

  = Hom R (B, R) t❡♠♦s q✉❡ ❛ s❡❣✉✐♥t❡ ❛♣❧✐❝❛çã♦ é ✉♠❛ ❜✐❥❡çã♦ d

  =1

  (ϕ ↼ ψ(x i ))(x j ) · ρ j ⊗ ρ i = n

  X i

  =1

  ϕ ↼ ψ(x i ) ⊗ ρ i = ϕ ⊗ n

  X i

  =1

  ψ(x i ) · ρ i ! = ϕ ⊗ ψ.

  ▲❡♠❛ ✶✳✹✹ ❙❡❥❛ B ✉♠ R✲❛♥❡❧✱ q✉❡ é ♣r♦❥❡t✐✈♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ❝♦♠♦ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❈♦♥s✐❞❡r❛♥❞♦ B

  ∗

  ( ) : B

  ϕ(ψ(x i ) · x j ) · ρ j ⊗ ρ i = n

  ∗

  ⊗ R B

  ∗

  ⊗ R B

  ∗

  → Hom R (B ⊗ R B ⊗ R

  B, R) ϕ ⊗ ψ ⊗ ξ 7→

  \ ϕ ⊗ ψ ⊗ ξ

  ❡♠ q✉❡ \

  ϕ ⊗ ψ ⊗ ξ(a ⊗ b ⊗ c) = ϕ(ψ(ξ(a) · b) · c) = ✱ ♣❛r❛ q✉❛✐sq✉❡r a, b

  X i,j

  =1

  X i,j

  =1

  =1

  F (x i ⊗ x j )ρ j (ρ i (a) · b) = n

  X i

  =1

  F  x i ⊗ n

  X j

  =1

  x j · ρ j (ρ i (a) · b)  

  = n

  X i

  F (x i ⊗ ρ i (a) · b) = F n

  X i,j

  X i

  =1

  x i · ρ i (a) ⊗ b !

  = F (a ⊗ b), ♣♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ q✉❛✐sq✉❡r ϕ, ψ ∈ B

  ∗

  ✱ t❡♠♦s ^ \

  ϕ ⊗ ψ = n

  X i,j

  =1

  \ ϕ ⊗ ψ(x i ⊗ x j ) · ρ j ⊗ ρ i

  = n

  ❡ c ∈ B✳

  ❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ ❞❡✜♥❛ g ( ) : Hom R (B ⊗ R B ⊗ R

  F (x i · ρ i (a) ⊗ b ⊗ c) = F (a ⊗ b ⊗ c),

  =1

  F (x i ⊗ x j ⊗ ρ j (ρ i (a) · b) · c) = n

  X i,j

  =1

  F (x i ⊗ x j · ρ j (ρ i (a) · b) ⊗ c) = n

  X i

  =1

  F (x i ⊗ ρ i (a) · b ⊗ c) = n

  X i

  =1

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ q✉❛✐sq✉❡r ϕ, ψ ξ ∈ B

  = n

  ∗

  ✱ t❡♠♦s ^ \

  ϕ ⊗ ψ ⊗ ξ = n

  X i,j,k

  =1

  \ ϕ ⊗ ψ ⊗ ξ(x i ⊗ x j ⊗ x k ) · ρ k ⊗ ρ j ⊗ ρ i

  = n

  X i,j,k =1 ϕ(ψ(ξ(x i ) · x j ) · x k ) · ρ k ⊗ ρ j ⊗ ρ i

  = n

  X i,j,k

  =1

  X i,j

  x k · ρ k (ρ j (ρ i (a) · b) · c) !

  B, R) −→ B

  B, R) ❡ a, b, c ∈ B✱ t❡♠♦s

  ∗

  ⊗ R B

  ∗

  ⊗ R B

  ∗

  ✱ F 7−→ e F t❛❧ q✉❡ e

  F := n

  X i,j,k =1 F (x i ⊗ x j ⊗ x k ) · ρ k ⊗ ρ j ⊗ ρ i ,

  ❡♠ q✉❡ x i , x j , x k ✱ ρ i , ρ j ❡ ρ k ❡stã♦ ♥❛ ❜❛s❡ ❞✉❛❧✳ ▼♦str❡♠♦s q✉❡ d ( )

  ❡ g ( ) sã♦ ✐♥✈❡rs❛s ✉♠❛ ❞❛ ♦✉tr❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r F ∈

  Hom R (B ⊗ R B ⊗ R

  F (a ⊗ b ⊗ c) = be \ 

  =1

   n

  X i,j,k

  =1

  F (x i ⊗ x j ⊗ x k ) · ρ k ⊗ ρ j ⊗ ρ i  (a ⊗ b ⊗ c)

  = n

  X i,j,k =1 F (x i ⊗ x j ⊗ x k )ρ k (ρ j (ρ i (a) · b) · c)

  = n

  X i,j

  =1

  F x i ⊗ x j ⊗ n

  X k

  (ϕ ↼ ψ(ξ(x i ) · x j ))(x k ) · ρ k ⊗ ρ j ⊗ ρ i n

  X = ϕ ↼ ψ(ξ(x i ) · x j ) ⊗ ρ j ⊗ ρ i i,j n =1

  X = ϕ ⊗ (ψ ↼ ξ(x i ))(x j ) · ρ j ⊗ ρ i i,j n =1

  X = ϕ ⊗ ψ ↼ ξ(x i ) ⊗ ρ i i

  =1 n

  X = ϕ ⊗ ψ ⊗ ξ(x i ) · ρ i i

  =1 = ϕ ⊗ ψ ⊗ ξ.

  Pr♦♣♦s✐çã♦ ✶✳✹✺ P❛r❛ ✉♠ R✲❛♥❡❧ (B, µ, η) ♣r♦❥❡t✐✈♦ ✜♥✐t❛♠❡♥t❡ ❣❡✲

  ∗

  r❛❞♦ ❝♦♠♦ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✱ ♦ ❞✉❛❧ à ❞✐r❡✐t❛ B ♣♦ss✉✐ ✉♠❛ ❡s✲ tr✉t✉r❛ ❝❛♥ô♥✐❝❛ ❞❡ R✲❝♦❛♥❡❧✳ ❖ ❝♦♣r♦❞✉t♦ ❡♠ t❡r♠♦s ❞❛ ❜❛s❡ ❞✉❛❧ n ∗ n ({x i ∈ B} , {ρ i ∈ B } ) i =1 i =1 ✱ é ❞❛❞♦ ♣♦r n

  X ∆(ϕ) = ϕ(x i −) ⊗ R ρ i . i

  =1 ∗ ∗

  −→ R B ) ❆ ❝♦✉♥✐❞❛❞❡ é ❞❛❞❛ ♣♦r ε : B ✱ ε(ϕ) = ϕ(1 ✱ ♣❛r❛ t♦❞♦ ϕ ∈ B ✳

  ∗

  ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ ❛ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦ ❡♠ B é

  ′ ∗ ′

  ∈ R )(b) = ❞❛❞❛✱ ♣❛r❛ q✉❛✐sq✉❡r r, r ✱ ϕ ∈ B ❡ b ∈ B✱ ♣♦r (r · ϕ ↼ r

  ′

  rϕ(r · b) ✳ ❆❣♦r❛ ♣❡❧♦ ▲❡♠❛ ✶✳✹✸ ✈❛♠♦s ✈❡r q✉❡ ♦ ❝♦♣r♦❞✉t♦ é ♠♦r✜s♠♦

  ∗

  ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R✱ ϕ ∈ B ❡ a, b ∈ B✱ t❡♠♦s n \ !

  X \

  ∆(ϕ ↼ r)(a ⊗ b) = (ϕ ↼ r)(x i −) ⊗ R ρ i (a ⊗ b) n i =1

  X = (ϕ ↼ r)(x i (ρ i (a) · b)) i =1 n

  X = (ϕ ↼ r)((x i · ρ i (a))b) i

  =1

  = (ϕ ↼ r)(ab) = ϕ(r · ab),

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ \

  ! (a ⊗ b)

  X i

  

=1

  ϕ((x i · ρ i (a))b) = rϕ(ab),

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ \

  (r · ∆(ϕ))(a ⊗ b) = \ r · n

  X i =1 ϕ(x i −) ⊗ R ρ i

  = \ n

  X i =1 (r · ϕ)(x i (ρ i (a) · b))

  X i

  =1

  r · ϕ(x i −) ⊗ R ρ i !

  (a ⊗ b) = n

  X i =1 rϕ(x i (ρ i (a) · b)) = r n

  X i

  =1

  = r n

  = n

  (∆(ϕ) ↼ r)(a ⊗ b) = \ n

  ϕ(x i ((ρ i ↼ r)(a) · b)) = n

  X i

  =1

  ϕ(x i −) ⊗ R ρ i ↼ r !

  (a ⊗ b) = n

  X i

  

=1

  X i =1 ϕ(x i (ρ i (r · a) · b))

  ! (a ⊗ b)

  = n

  X i

  

=1

  ϕ((x i · ρ i (r · a))b) = ϕ((r · a)b) = ϕ(r · ab).

  ❚❛♠❜é♠ t❡♠♦s \

  ∆(r · ϕ)(a ⊗ b) = \ n

  X i =1 (r · ϕ)(x i −) ⊗ R ρ i

  ϕ((x i · ρ i (a))b)

  = rϕ(ab). ❙❡❣✉❡ q✉❡ ∆ é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❆❣♦r❛ ✉s❛♥❞♦ ♦ ▲❡♠❛ ✶✳✹✹✱ ✈❛♠♦s ♠♦str❛r q✉❡ ✈❛❧❡ ❛ ❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r

  ∗

  a, b, c ∈ B ❡ ϕ ∈ B ✱ t❡♠♦s

  \

  ∗

  ((B ⊗ R ∆)∆(ϕ))(a ⊗ b ⊗ c) = n \ !!

  X

  ∗

  = (B ⊗ ∆) ϕ(x i −) ⊗ R ρ i (a ⊗ b ⊗ c) i

  =1

  \   n

  X  (a ⊗ b ⊗ c)

  = ϕ(x i −) ⊗ R ρ i (x j −) ⊗ R ρ j i,j n =1

  X = ϕ(x i (ρ i (x j (ρ j (a) · b)) · c)) i,j n =1

  X = ϕ(x i (ρ i ((x j · ρ j (a))b)) · c) i,j n =1

  X = ϕ(x i (ρ i (ab) · c)) i

  =1 n

  X = ϕ((x i · ρ i (ab))c)) i

  =1

  = ϕ(abc), ♣♦r ♦✉tr♦ ❧❛❞♦✱

  \

  ∗

  ((∆ ⊗ R B )∆(ϕ))(a ⊗ b ⊗ c) = \ !! n

  X

  ∗

  = (∆ ⊗ B ) ϕ(x i −) ⊗ R ρ i (a ⊗ b ⊗ c) i

  =1

   \  n

  X  (a ⊗ b ⊗ c)

  = ϕ(x i x j −) ⊗ R ρ j ⊗ R ρ i i,j n =1

  X = ϕ(x i x j (ρ j (ρ i (a) · b) · c)) i,j n =1

  X = ϕ(x i (x j · ρ j (ρ i (a) · b))c) i,j

  =1 n

  X = ϕ(x i (ρ i (a) · b)c) i

  =1 n

  X = ϕ((x i · ρ i (a))bc) i

  =1 = ϕ(abc).

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ✈❛❧❡ ❛ ❝♦✉♥✐t❛❧✐❞❛❞❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r

  ∗

  ϕ ∈ B ❡ b ∈ B✱ t❡♠♦s n !

  X

  ∗

  ((B ⊗ R ε)∆(ϕ)) (b) = ϕ(x i −) ↼ ε(ρ i ) (b) n i =1

  X = ϕ(x i (ρ i (1 B ) · b)) i

  

=1

n

  X = ϕ((x i · ρ i (1 B ))b) i

  

=1

  = ϕ(1 B

  b) = ϕ(b) ❡ n

  X

  ∗

  (ε ⊗ R B )∆(ϕ) = ε(ϕ(x i −)) · ρ i i =1 n

  X = ϕ(x i ) · ρ i i

  =1 = ϕ.

  ◆♦ss♦ ♣ró①✐♠♦ ❧❡♠❛ ♠♦str❛ ❛ ❝♦♥❞✐çã♦✱ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡✱ ♣❛r❛ R

  ❝♦♠♦ ♠ó❞✉❧♦ r❡❣✉❧❛r ✭R é ✉♠ ♠ó❞✉❧♦ s♦❜r❡ ❡❧❡ ♠❡s♠♦✮✱ s❡r ✉♠ A

  ✲♠ó❞✉❧♦✱ ❡♠ q✉❡ A é ✉♠ R✲❛♥❡❧✳ ❊st❡ ❧❡♠❛ t❛♠❜é♠ s❡rá ✉s❛❞♦ q✉❛♥❞♦ ❡st✐✈❡r♠♦s tr❛t❛♥❞♦ ❞❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❞❡ ♠ó❞✉❧♦s s♦❜r❡ ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛ ♥♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✳ ❊♠ t❛❧ ❧❡♠❛✱ t❛♠❜é♠ ❛♣❛r❡❝❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝❛r❛❝t❡r à ❞✐r❡✐t❛✱ q✉❡ s❡rá ✉s❛❞♦ ♥❛ ❞❡✜♥✐çã♦ ❞❡ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ▲❡♠❛ ✶✳✹✻ ❖ ♠ó❞✉❧♦ à ❞✐r❡t❛ r❡❣✉❧❛r R s❡ ❡st❡♥❞❡ ♣❛r❛ ✉♠ ♠ó❞✉❧♦ à ❞✐r❡✐t❛ s♦❜r❡ ✉♠ R✲❛♥❡❧ (A, µ, η) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠ ♠♦r✜s♠♦ ❞❡ k✲♠ó❞✉❧♦ χ : A −→ R✱ t❛❧ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ A ❡ r ∈ R ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s✳

  A ) = 1 R

  ✭✐✮ χ(1 ❀ ✭✐✐✮ χ(aη(r)) = χ(a)r❀

  ✭✐✐✐✮ χ((η ◦ χ)(a)b) = χ(ab)✳ ❉❡♠♦♥str❛çã♦✿ (⇒) ◗✉❡r❡♠♦s ❞❡✜♥✐r χ : A −→ R ♠♦r✜s♠♦ ❞❡ k✲ R : R ⊗ R A −→ R ♠ó❞✉❧♦s✳ ❈♦♠♦ R é A✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✱ t❡♠♦s ψ ❡♠ R (r ⊗ a) = r ↼ a q✉❡ ψ é ❛ ❛çã♦ à ❞✐r❡✐t❛ ❞❡ A ❡♠ R✳ ❚❛♠❜é♠ t❡♠♦s l A 1 ψ R A // R ⊗ R A // R . R ◦ l −1

  ❉❡✜♥❛ χ := ψ A ✱ ❞❡ss❛ ❢♦r♠❛ t❡♠♦s

  −1

  χ(a) = (ψ R ◦ l )(a) = ψ R (1 R ⊗ R A a) = 1 R ↼ a. ◆♦t❡ q✉❡ χ é ♠♦r✜s♠♦ ❞❡ k✲♠ó❞✉❧♦s✱ ♣♦✐s é ❛ ❝♦♠♣♦s✐çã♦ ❞❡ ♠♦r✜s♠♦s ❞❡ k✲♠ó❞✉❧♦s✳ ❱❛♠♦s ✈❡r✐✜❝❛r ❛s ♦✉tr❛s ❝♦♥❞✐çõ❡s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ A ) = 1 R ↼ 1 A = 1 R q✉❛✐sq✉❡r a, b ∈ A ❡ r ∈ R✱ t❡♠♦s χ(1 ❡ t❛♠❜é♠

  χ(aη(r)) = 1 R ↼ (aη(r)) = (1 R ↼ a) ↼ η(r) = (1 R ↼ a)r = χ(a)r,

  ❡ ❛✐♥❞❛ χ((η ◦ χ)(a)b) = χ(η(χ(a))b)

  = 1 A ↼ η(χ(a))b = (1 R ↼ η(χ(a))) ↼ b = (1 R ↼ 1 A · χ(a)) ↼ b = ((1 R ↼ 1 A )χ(a)) ↼ b = χ(a) ↼ b = (1 R ↼ a) ↼ b = 1 R ↼ ab = χ(ab).

  (⇐) ❚❡♠♦s χ : A −→ R ♠♦r✜s♠♦ ❞❡ k✲♠ó❞✉❧♦s q✉❡ s❛t✐s❢❛③ ❛s

  ❝♦♥❞✐çõ❡s ❛❝✐♠❛✳ ❉❡✜♥❛✱ f ψ R : R × A → R (r, a) 7→ χ(η(r)a).

  ψ R ▼♦str❡♠♦s ♣r✐♠❡✐r♦ q✉❡ f é R✲❜❛❧❛♥❝❡❛❞❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r, s ∈ R

  ❡ a ∈ A t❡♠♦s f ψ R (r, s · a) =χ(η(r)(s · a))

  = χ(η(r)η(s)a) = χ(η(rs)a) = f ψ R (rs, a).

  ψ R R : R⊗ R A −→ R ❙❡❣✉❡ q✉❡ f é R✲❜❛❧❛♥❝❡❛❞❛✳ P♦rt❛♥t♦✱ ❡①✐st❡ ú♥✐❝❛ ψ ✱ R (r ⊗ a) = χ(η(r)a) t❛❧ q✉❡ ψ ✱ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R ❡ a ∈ A✳ ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s ψ R R ⊗A ⊗η

  // // R ⊗ R A ⊗ R A R ⊗ R A R ∼ = R ⊗ R R R ⊗ R A R ⊗µ ψ R R Id R ψ

  '' R. R ⊗ R A // R, ψ R

  ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ A ❡ r ∈ R✱ t❡♠♦s ♣❛r❛ ♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛

  (ψ R ◦ (ψ R ⊗ A))(r ⊗ a ⊗ b) = ψ R (χ(η(r)a) ⊗ b) = χ(η(χ(η(r)a))b) = χ(η(r)ab) = ψ R (r ⊗ ab) = (ψ R ◦ (R ⊗ µ))(r ⊗ a ⊗ b).

P❛r❛ ♦ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛ t❡♠♦s

  (ψ R ◦ (R ⊗ η))(r ⊗ 1 R ) = ψ R (r ⊗ η(1 R )) = ψ R (r ⊗ 1 A ) = χ(η(r)1 A ) = χ(1 A η(r)) = χ(1 A )r = 1 R r = r.

  P♦rt❛♥t♦✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳ ❉❡✜♥✐çã♦ ✶✳✹✼ ❙❡❥❛ (A, µ, η) ✉♠ R✲❛♥❡❧✳ ❯♠ ♠♦r✜s♠♦ ❞❡ k✲♠ó❞✉❧♦s χ : A −→ R

  ✱ s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❞♦ ❧❡♠❛ ❛❝✐♠❛ é ❝❤❛♠❛❞♦ ❞❡ ❝❛r❛❝t❡r à ❞✐r❡✐t❛✳

  ❉❡✜♥✐çã♦ ✶✳✹✽ ❙❡❥❛ (A, µ, η) ✉♠ R✲❛♥❡❧ ♣♦ss✉✐♥❞♦ ✉♠ ❝❛r❛❝t❡r à ❞✐✲ r❡✐t❛ χ : A −→ R✳ ❖s ✐♥✈❛r✐❛♥t❡s ❞❡ ✉♠ A✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ (M, ·) ❝♦♠ r❡s♣❡✐t♦ ❛♦ ❝❛r❛❝t❡r χ sã♦ ♦s ❡❧❡♠❡♥t♦s ❞♦ k✲s✉❜♠ó❞✉❧♦ M χ := {m ∈ M ; m · a = m · η(χ(a)), ∀a ∈ A} .

  Pr♦♣♦s✐çã♦ ✶✳✹✾ ❙❡❥❛♠ (A, µ, η) ✉♠ R✲❛♥❡❧ ❝♦♠ ✉♠ ❝❛r❛❝t❡r à ❞✐✲ χ r❡✐t❛ χ : A −→ R ❡ M ✉♠ A✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❊♥tã♦ M é ✐s♦♠♦r❢♦ A (R, M ) ❛ Hom ❝♦♠♦ k✲♠ó❞✉❧♦s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦s ✐♥✈❛r✐❛♥t❡s ❞❡ R sã♦ ♦s ❡❧❡♠❡♥t♦s ❞❛ s✉❜á❧❣❡❜r❛

  B = {r ∈ R; χ(η(r)a) = rχ(a), ∀ a ∈ A} . χ −→ Hom A (R, M ) ❉❡♠♦♥str❛çã♦✿ ❉❡✜♥❛ ϕ : M ✱ ϕ(m)(r) = m·η(r)✱ χ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M ❡ r ∈ R✳ ▼♦str❡♠♦s q✉❡ ϕ(m) é ♠♦r✜s♠♦ ❞❡ A R (r ⊗ a) = r ⊳ a = χ(η(r)a)

  ✲♠ó❞✉❧♦s à ❞✐r❡✐t❛✳ ❉❡ ❢❛t♦✱ ❞❡♥♦t❛♠♦s ψ χ ❡ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M ✱ a ∈ A ❡ r ∈ R✱ t❡♠♦s

  ϕ(m)(r ⊳ a) = m · η(r ⊳ a) = m · η(χ(η(r)a)) = m · η(r)a = (m · η(r)) · a = ϕ(m)(r) · a.

  ➱ ❢á❝✐❧ ✈❡r q✉❡ ϕ é ♠♦r✜s♠♦ ❞❡ k✲♠ó❞✉❧♦s✳ ❆❣♦r❛ s❡❥❛ m ∈ ker(ϕ)✱ ❡♥tã♦ m · η(r) = ϕ(m)(r) = 0 ♣❛r❛ t♦❞♦ r ∈ R✱ ❡♠ ♣❛rt✐❝✉❧❛r ♣❛r❛ r = 1 R s❡❣✉❡ q✉❡ m = 0✱ ♦✉ s❡❥❛✱ ϕ é ✐♥❥❡t♦r❛✳ ❆❣♦r❛ ❝♦♥s✐❞❡r❡ f ∈

  Hom A (R, M ) R ) ∈ M R ) ∈ M χ q✉❛❧q✉❡r✱ ❞❡✜♥❛ m = f(1 ❡ ♥♦t❡ q✉❡ f(1 ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ a ∈ A✱ t❡♠♦s f (1 R ) · η(χ(a)) = f (1 R ⊳ η(χ(a)))

  = f (χ(η(1 R )η(χ(a)))) = f (χ(η(χ(a)))) = f (χ(a)) = f (χ(η(1 R )a)) = f (1 R ⊳ a) = f (1 R ) · a.

  ❆ss✐♠ ♣❛r❛ t♦❞♦ r ∈ R✱ t❡♠♦s ϕ(f (1 R ))(r) = f (1 R ) · η(r)

  = f (1 R ⊳ η(r)) = f (χ(η(1 R )η(r))) = f (χ(1 A )r) = f (1 R r) = f (r).

  ❙❡❣✉❡ ❞❛í q✉❡ ϕ é s♦❜r❡❥❡t♦r❛✳ P♦rt❛♥t♦✱ ϕ é ✐s♦♠♦r✜s♠♦✳ ▼♦str❡♠♦s χ χ ❛❣♦r❛ q✉❡ B = R ✳ P❛r❛ t❛♥t♦✱ ♠♦str❡♠♦s ♣r✐♠❡✐r♦ q✉❡ B ⊆ R ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ r ∈ B✱ t❡♠♦s r ⊳ η(χ(a)) = χ(η(r)η(χ(a)))

  = χ(η(r))χ(a) = χ(1 A )rχ(a) = rχ(a) = χ(η(r)a) χ χ ⊆ B = r ⊳ a,

  ♦✉ s❡❥❛✱ r ∈ R ✳ ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ R ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ i ∈ R χ ✱ t❡♠♦s

  χ(η(r)a) = r ⊳ a = r ⊳ η(χ(a)) = χ(η(r)η(χ(a))) = χ(η(r))χ(a) = χ(1 A )rχ(a) = rχ(a), χ

  ♦✉ s❡❥❛✱ r ∈ B✳ ❙❡❣✉❡ ♣♦rt❛♥t♦✱ q✉❡ B = R ✳ ❆ss♦❝✐❛❞♦ ❛♦ ❝❛r❛❝t❡r χ✱ ❡①✐st❡ ✉♠ ♠♦r✜s♠♦ ❝❛♥ô♥✐❝♦

  F : A → B End(R) χ a 7→ r 7−→ χ(η(r)a), ❡♠ q✉❡ B = R ❞❡✜♥✐❞♦ ♥❛ ♣r♦♣♦s✐çã♦ ✶✳✹✾✳ ❉❡✜♥✐çã♦ ✶✳✺✵ ❖ R✲❛♥❡❧ (A, µ, η) é ❞✐t♦ s❡r ❞❡ ●❛❧♦✐s ❝♦♠ r❡s♣❡✐t♦ ❛ χ s❡ F é ❜✐❥❡t✐✈❛✳ ▲❡♠❛ ✶✳✺✶ ❖ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ r❡❣✉❧❛r R s❡ ❡st❡♥❞❡ ♣❛r❛ ✉♠ C✲ ❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❞❡ ✉♠ R✲❝♦❛♥❡❧ (C, ∆, ε) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠ ❣r♦✉♣✲❧✐❦❡ g ∈ C✳

  ❉❡♠♦♥str❛çã♦✿ (=⇒) R é C✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛✱ ♦✉ s❡❥❛✱ ♦s ❞✐❛❣r❛✲ ♠❛s s❡❣✉✐♥t❡s ❝♦♠✉t❛♠ ρ ρ

  // ρ R // R ⊗ C R R ⊗ C . R R

  ⊗∆ ⊗ε l R 1

  %% R ⊗ C // R ⊗ C ⊗ C, R ∼ = R ⊗ R. ρ

  ⊗C R )

  ❆❣♦r❛ ❞❡✜♥❛ g = ρ(1 ✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡ ✉t✐❧✐③❛♥❞♦ ♦ ✐s♦♠♦r✜s♠♦ l C : R ⊗ C −→ C c ◦ ρ(1 R ). t❡♠♦s g = l ❆ss✐♠✱ l C (R ⊗ ∆)(1 R ⊗ g) = l C (1 R ⊗ g ⊗ g ) = g ⊗ g = ∆(g).

  ⊗C ⊗C (1) (2) (1) (2)

  P♦rt❛♥t♦✱ ∆(g) = l C (R ⊗ ∆)(1 R ⊗ g)

  ⊗C

  = l C (R ⊗ ∆)ρ(1 R )

  ⊗C

  = l C ⊗C (ρ ⊗ C)ρ(1 R ) = (l C ⊗ C)(ρ ⊗ C)(1 R ⊗ g) = l C ◦ ρ(1 R ) ⊗ g = g ⊗ g.

  ❚❛♠❜é♠ t❡♠♦s

  −1

  1 R = l R ◦ l (1 R ) R = l R (R ⊗ ε)ρ(1 R ) = l R (R ⊗ ε)(1 R ⊗ g) = l R (1 R ⊗ ε(g)) = ε(g).

  (⇐=) ❆❣♦r❛✱ s❡❥❛ g ∈ C✱ s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❛❝✐♠❛✱ ❞❡✜♥❛

  ρ : R −→ R ⊗ C r 7−→

  1 R ⊗ g · r, ♥♦t❡ q✉❡ ρ é ♠♦r✜s♠♦ ❞❡ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❉❡ ❢❛t♦✱

  ρ(rs) = 1 R ⊗ g · (rs) = 1 R ⊗ (g · r) · s = (1 R ⊗ g · r)s = ρ(r)s. P♦rt❛♥t♦✱ ρ é ♠♦r✜s♠♦ ❞❡ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❱❛♠♦s ♠♦str❛r q✉❡ ρ é ✉♠❛ ❝♦❛çã♦ à ❞✐r❡✐t❛✳ ❉❡ ❢❛t♦✱ t❡♠♦s

  (R ⊗ ∆)ρ(r) = (R ⊗ ∆)(1 R ⊗ g · r) = 1 R ⊗ ∆(g · r) = 1 R ⊗ ∆(g)r = 1 R ⊗ (g ⊗ g)r = 1 R ⊗ g ⊗ g · r = ρ(1 R ) ⊗ g · r = (ρ ⊗ C)(1 R ⊗ g · r) = (ρ ⊗ C)ρ(r).

  ❡ (R ⊗ ε)ρ(r) = (R ⊗ ε)(1 R ⊗ g · r)

  = 1 R ⊗ ε(g · r) = 1 R ⊗ ε(g)r

  −1 = 1 R ⊗ r = l (r). R

  ❙❡❣✉❡ ♣♦rt❛♥t♦✱ q✉❡ R t❡♠ ❡str✉t✉r❛ ❞❡ C✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❉❡st❡ ♣♦♥t♦ ❡♠ ❞✐❛♥t❡✱ ❢❛r❡♠♦s ❛s ✐❞❡♥t✐✜❝❛çõ❡s s❡♠♣r❡ q✉❡ ❛s M M

  ❡①♣r❡ssõ❡s ❡♥✈♦❧✈❡r❡♠ ♦s ✐s♦♠♦r✜s♠♦s l ❡ r ✳ ❉❡✜♥✐çã♦ ✶✳✺✷ ❙❡❥❛ (C, ∆, ε) ✉♠ R✲❝♦❛♥❡❧ ♣♦ss✉✐♥❞♦ ✉♠ ❣r♦✉♣❧✐❦❡ g ∈ C M )

  ✳ ❖s ❝♦✐♥✈❛r✐❛♥t❡s ❞❡ ✉♠ ❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛ (M, ρ ❝♦♠ r❡s♣❡✐t♦ ❛ g✱ sã♦ ♦s ❡❧❡♠❡♥t♦s ❞♦ k✲s✉❜♠ó❞✉❧♦ g M := {m ∈ M ; ρ M (m) = m ⊗ g} .

  Pr♦♣♦s✐çã♦ ✶✳✺✸ ❙❡❥❛♠ (C, ∆, ε) ✉♠ R✲❝♦❛♥❡❧ ❝♦♠ ✉♠ ❣r♦✉♣❧✐❦❡ g ∈ g C C

  (R, M ) ❡ M ✉♠ C✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❊♥tã♦ M é ✐s♦♠♦r❢♦ ❛ Hom

  ❝♦♠♦ k✲♠ó❞✉❧♦s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦s ❝♦✐♥✈❛r✐❛♥t❡s ❞❡ R sã♦ ♦s ❡❧❡♠❡♥✲ t♦s ❞❛ s✉❜á❧❣❡❜r❛ B = {r ∈ R; r · g = g · r} . g C

  −→ Hom (R, M ) ❉❡♠♦♥str❛çã♦✿ ❉❡✜♥❛ ϕ : M ✱ ϕ(m)(r) = m · r ♣❛r❛ t♦❞♦ r ∈ R✳ ❱❛♠♦s ✈❡r q✉❡ ϕ(m) é ♠♦r✜s♠♦ ❞❡ C✲❝♦♠ó❞✉❧♦s à g ❞✐r❡✐t❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M ❡ r ∈ R✱ t❡♠♦s

  ρ M (ϕ(m)(r)) = ρ M (m · r)

  = ρ M (m) · r = m ⊗ g · r = m · 1 R ⊗ g · r = ϕ(m)(1 R ) ⊗ g · r = (ϕ(m) ⊗ C)(1 R ⊗ g · r) C g = (ϕ(m) ⊗ C)ρ R (r).

  (R, M ) −→ M R ) ❆❣♦r❛ ❞❡✜♥❛ ψ : Hom t❛❧ q✉❡ ψ(f) = f(1 ♣❛r❛ C g

  (R, M ) R ) ∈ M t♦❞♦ f ∈ Hom ✳ ❱❛♠♦s ✈❡r q✉❡ f(1 ✳ ❉❡ ❢❛t♦✱ t❡♠♦s ρ M (f (1 R )) = (f ⊗ C)ρ R (1 R )

  = (f ⊗ C)(1 R ⊗ g · 1 R ) = f (1 R ) ⊗ g. ▼♦str❡♠♦s q✉❡ ϕ ❡ ψ sã♦ ✐♥✈❡rs❛s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R✱ g C m ∈ M (R, M )

  ❡ f ∈ Hom t❡♠♦s (ϕ ◦ ψ)(f )(r) = ϕ(f (1 R ))(r)

  = f (1 R ) · r = f (1 R · r) = f (r).

  ❚❛♠❜é♠ t❡♠♦s (ψ ◦ ϕ)(m) = ψ(ϕ(m))

  = ϕ(m)(1 R ) = m · 1 R g = m.

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ B = R ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ r ∈ B✱ t❡♠♦s ρ R (r) = 1 R ⊗ g · r

  = 1 R ⊗ r · g g = r ⊗ g.

P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ t♦❞♦ r ∈ R ✱ t❡♠♦s

  1 R ⊗ g · r = ρ R (r) = r ⊗ g

  = 1 R ⊗ r · g. g ❙❡❣✉❡ ♣♦rt❛♥t♦✱ q✉❡ r · g = g · r✳ ▲♦❣♦ B = R ✳

  ❆ss♦❝✐❛❞♦ ❛♦ ❣r♦✉♣❧✐❦❡ g✱ ❡①✐st❡ ✉♠ ♠♦r✜s♠♦ ❝❛♥ô♥✐❝♦✱ Can : R ⊗ R −→ C a ⊗ b 7−→ a · g · b.

  ❉❡✜♥✐çã♦ ✶✳✺✹ ❖ R✲❝♦❛♥❡❧ C é ❞✐t♦ s❡r ✉♠ R✲❝♦❛♥❡❧ ❞❡ ●❛❧♦✐s s❡ Can

  é ✉♠❛ ❜✐❥❡çã♦✳ Pr♦♣♦s✐çã♦ ✶✳✺✺ ❙❡❥❛ C ✉♠ R✲❝♦❛♥❡❧ ♣r♦❥❡t✐✈♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ g : C −→

  ❝♦♠♦ R✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛✳ P❛r❛ g ∈ C✱ t❡♠♦s ❛ ❛♣❧✐❝❛çã♦ χ R

  ✱ φ 7−→ φ(g)✳ ❊♥tã♦ ✈❛❧❡ ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿ g ✭✶✮ ❖ ❡❧❡♠❡♥t♦ g ∈ C é ❣r♦✉♣❧✐❦❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ χ é ✉♠ ❝❛r❛❝t❡r

  ∗

  C à ❞✐r❡✐t❛ ♥♦ R✲❛♥❡❧ ❀

  ✭✷✮ ❯♠ ❡❧❡♠❡♥t♦ b ∈ R é ✉♠ ❝♦✐♥✈❛r✐❛♥t❡ ❞♦ C✲❝♦♠ó❞✉❧♦ R à ❞✐r❡✐t❛ ✭❝♦♠ ❝♦❛çã♦ ✐♥❞✉③✐❞❛ ♣♦r ✉♠ ❡❧❡♠❡♥t♦ ❣r♦✉♣❧✐❦❡ ❣✮ s❡✱ ❡ s♦♠❡♥t❡

  ∗

  C s❡✱ b é ✉♠ ✐♥✈❛r✐❛♥t❡ ❞♦ ✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ R ✭❝♦♠ r❡s♣❡✐t♦ ❛♦ g ❝❛r❛❝t❡r à ❞✐r❡✐t❛ χ ✮❀

  ✭✸✮ ❙❡ ♦ R✲❝♦❛♥❡❧ C é ✉♠ ❝♦❛♥❡❧ ❞❡ ●❛❧♦✐s ✭❝♦♠ r❡s♣❡✐t♦ ❛♦ ❡❧❡♠❡♥t♦

  

  C ❣r♦✉♣❧✐❦❡ ❣✮✱ ❡♥tã♦ ♦ R✲❛♥❡❧ é ✉♠ ❛♥❡❧ ●❛❧♦✐s ✭❝♦♠ r❡s♣❡✐t♦ g ❛♦ ❝❛r❛❝t❡r à ❞✐r❡✐t❛ χ ✮✳

  ❉❡♠♦♥str❛çã♦✿ ❆♥t❡s ❞❡ ♣r♦✈❛r♠♦s ❛ ♣r♦♣♦s✐çã♦ ✈❛♠♦s r❡❧❡♠❜r❛r

  

  C q✉❡✱ ♦ ♣r♦❞✉t♦ ❞❡ ❝♦♥✈♦❧✉çã♦ ❡♠ ✭✈❡r Pr♦♣♦s✐çã♦ ✶✳✹✷✮ é ❞❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

  (φ ∗ ψ)(g) = ψ(g · φ(g ))

  (1)

  2 ∗ ∗

  C C ♣❛r❛ q✉❛✐sq✉❡r g ∈ C ❡ φ, ψ ∈ ✳ ❊ ❛✐♥❞❛ η : R −→ é ❞❡✜♥✐❞♦ ♣♦r η(r)(g) = ε(g)r ♣❛r❛ q✉❛✐sq✉❡r g ∈ C ❡ r ∈ R✳

  ✭✶✮ (=⇒) ❚❡♠♦s q✉❡ ✈❡r✐✜❝❛r ❛s ❝♦♥❞✐çõ❡s ✭✐✮✱ ✭✐✐✮ ❡ ✭✐✐✐✮ ❞♦ ▲❡♠❛

  ∗

  C ✶✳✹✻✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r φ, ψ ∈ ✱ r ∈ R ❡ g ∈ C ✉♠ ❣r♦✉♣❧✐❦❡✱ t❡♠♦s✱ g (1 C ) = 1 R C = ε ∗ ∗

  ✭✐✮ χ ✱ ❡♠ q✉❡ 1 ✳ ❉❡ ❢❛t♦✱ χ g (ε) = ε(g) = 1 R ;

  ✭✐✐✮ χ g (φ ∗ η(r)) = χ g (φ)r ✳ ❉❡ ❢❛t♦✱ t❡♠♦s χ g (φ ∗ η(r)) = (φ ∗ η(r))(g)

  (2)

  (2) )).

  · φ(g

  (1)

  ψ(g · φ(g)) = ψ(g

  C ✱ t❡♠♦s

  ∗

  ❙❡❣✉❡ ♣♦rt❛♥t♦✱ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r φ, ψ ∈

  (2) )).

  · φ(g

  (1)

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ χ g (φ ∗ ψ) = (φ ∗ ψ)(g) = ψ(g

  )χ g (φ)) = ψ(g · χ g (φ)) = ψ(g · φ(g)),

  · ε(g

  = η(r)(g · (φ(g))) = ε(g · (φ(g)))r = ε(g)φ(r)r = φ(g)r = χ g (φ)r;

  (1)

  )) = ψ(g

  (2)

  · η(χ g (φ))(g

  (1)

  = ψ(g

  ❆❣♦r❛ ♥♦t❡ q✉❡✱ χ g (η(χ g (φ)) ∗ ψ) = (η(χ g (φ)) ∗ ψ)(g)

  C ✳ ❉❡ ❢❛t♦✱ t❡♠♦s ε(g) = χ g (ε) = 1 R .

  ∗

  à ❞✐r❡✐t❛ ❡♠

  (⇐=) ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ g é ❣r♦✉♣❧✐❦❡✱ s❛❜❡♥❞♦ q✉❡ χ g é ❝❛r❛❝t❡r

  = ψ(g · η(χ g (φ))(g)) = ψ(g · ε(g)χ g (φ)) = ψ(g · χ g (φ)) = ψ(g · φ(g)) = (φ ∗ ψ)(g) = χ g (φ ∗ ψ).

  ✭✐✐✐✮ χ g (η(χ g (φ)) ∗ ψ) = χ g (φ ∗ ψ) ✳ ❉❡ ❢❛t♦✱ χ g (η(χ g (φ)) ∗ ψ) = (η(χ g (φ)) ∗ ψ)(g)

  ✭✶✳✸✮

  ❆❣♦r❛ ❝♦♠♦ C é ♣r♦❥❡t✐✈♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ❝♦♠♦ R✲♠ó❞✉❧♦ à ❡s✲ q✉❡r❞❛✱ t❡♠♦s {f i

  f i (g

  )) · x j ⊗ x i ♣♦r ✶✳✸

  = n

  X i =1 g

  (1)

  · f i (g

  (2)

  ) ⊗ x i = g

  (1)

  ⊗ n

  X i

  =1

  (2)

  · f i (g

  ) · x i = g

  (1)

  ⊗ g

  (2) .

  P♦rt❛♥t♦✱ ∆(g) = g (1) ⊗ g

  (2)

  = g ⊗ g ✳ ❙❡❣✉❡ q✉❡ g é ❣r♦✉♣❧✐❦❡✳

  ✭✷✮ (=⇒) b é ❝♦✐♥✈❛r✐❛♥t❡ ❞❡ R✱ ♦✉ s❡❥❛✱ b · g = g · b✳ ❚❡♠♦s q✉❡ ♠♦str❛r q✉❡ χ g (η(b) ∗ φ) = b χ g (φ).

  ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ φ ∈

  ∗

  C ✱ t❡♠♦s

  (2)

  (1)

  ∗

  X i

  C} n i

  =1

  ❡ {x i ∈ C} n i

  =1

  ❛ ❜❛s❡ ❞✉❛❧ ❞❡ C✱ ♦✉ s❡❥❛✱ ♣❛r❛ g ∈ C✱ t❡♠♦s g = n

  X i

  =1

  f i (g) · x i ❡ ♣❛r❛ g · f i

  (g) ✱ t❡♠♦s g · f i (g) = n

  X j =1 f j (g · f i (g)) · x j .

  ❉❡st❛ ❢♦r♠❛✱ g ⊗ g = g ⊗ n

  =1

  f j (g

  f i (g) · x i = n

  X i =1 g · f i (g) ⊗ x i = n

  X i

  =1 n

  X j

  =1

  f j (g · f i (g)) · x j ⊗ x i = n

  X i

  =1 d

  X j

  =1

  χ g (η(b) ∗ φ) = (η(b) ∗ φ)(g) = φ(g · η(b)(g))

  = φ(g · ε(g)b) = φ(g · b) = φ(b · g) = bφ(g) = b χ g (φ).

  φ : B End(R) −→

  ❆❣♦r❛ ❝♦♠♦ Can : R ⊗ B R −→ C ✱ r ⊗ r

  ′

  7−→ r · g · r

  ′

  ✱ é ❜✐❥❡t✐✈❛✱ t❡♠♦s q✉❡ ♣❛r❛ t♦❞♦ c ∈ C✱ ❡①✐st❡♠ P a i ⊗ b i ∈ R ⊗ B R t❛✐s q✉❡

  P a i · g · b i = c ✳ ❊♥tã♦ ❞❡✜♥❛

  ∗

  C ✱ t❡♠♦s

  C f 7−→ φ(f ), t❛❧ q✉❡✱ φ(f)(c) =

  P a i f (b i ) ✳ ❉❡st❛ ❢♦r♠❛✱ ♣❛r❛ q✉❛✐sq✉❡r

  P a i · g · b i = c ∈ C

  ✱ ϕ ∈

  ∗

  C ✱ f ∈ B End(R) ❡ r ∈ R✱ t❡♠♦s

  F (ϕ)(r) = χ g (η(r) ∗ ϕ) = (η(r) ∗ ϕ)(g) = ϕ(g · η(r)(g)) = ϕ(g · ε(g)r) = ϕ(g · r).

  

  P♦rt❛♥t♦✱ b é ✐♥✈❛r✐❛♥t❡ ❞♦

  X i

  ∗

  C ✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ R ✭❝♦♠ r❡s♣❡✐t♦ ❛ χ g ✮✳

  (⇐=) ❚❡♠♦s q✉❡ b é ✐♥✈❛r✐❛♥t❡ ❞❡ R✱ ♦✉ s❡❥❛✱ χ g (η(b) ∗ φ) = b χ g (φ).

  ❆ss✐♠✱ ♣❛r❛ t♦❞♦ φ ∈

  ∗

  C t❡♠♦s φ(g · b) = φ(b · g)✳ ❚❡♠♦s q✉❡ ♠♦str❛r q✉❡ b · g = g · b✳ ❉❡ ❢❛t♦✱ ✉s❛♥❞♦ ♥♦✈❛♠❡♥t❡ ❛ ❜❛s❡ ❞✉❛❧ ❞❡ C✱ t❡♠♦s b · g = n

  =1

  ♣❛r❛ q✉❛✐sq✉❡r r ∈ R ❡ ϕ ∈

  f i (b · g) · x i . ❉❡st❛ ❢♦r♠❛✱ b · g = n

  X i =1 f i (b · g) · x i = n

  X i =1 f i (g · b) · x i = g · b.

  P♦rt❛♥t♦✱ b é ❝♦✐♥✈❛r✐❛♥t❡ ❞♦ C✲❝♦♠ó❞✉❧♦ R ✭❝♦♠ r❡s♣❡✐t♦ ❛ g✮✳ ✭✸✮ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ F :

  

  C −→ B End(R) é ❜✐❥❡t✐✈❛✱ t❛❧ q✉❡✱

  (F ◦ φ)(f ) = F (φ(f ))(r)

  = φ(f )(g · r) = f (r),

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ ((φ ◦ F )(ϕ))(c) = φ(F (ϕ))(r)

  X = a i F (ϕ)(b i )

  X = a i ϕ(g · b i )

  X = ϕ( a i · g · b i ) = ϕ(c).

  ❙❡❣✉❡ q✉❡ F é ❜✐❥❡t✐✈❛✳

  ❈❛♣ít✉❧♦ ✷ ❇✐❛❧❣❡❜ró✐❞❡s

  ❯♠ ❜✐❛❧❣❡❜ró✐❞❡✱ ❞✐❢❡r❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❜✐á❧❣❡❜r❛s✱ ♥ã♦ é ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠❛ ❝♦♠♣❛t✐❜✐❞❛❞❡ ❡♥tr❡ ♠♦♥ó✐❞❡s ✭♦❜❥❡t♦s á❧❣❡❜r❛s✮ ❡ ❝♦♠♦♥ó✐✲ ❞❡s ✭♦❜❥❡t♦s ❝♦á❧❣❡❜r❛s✮ ♥❛ ♠❡s♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❞❡ ❜✐♠ó❞✉❧♦s✳ ❆s ❡str✉t✉r❛s ❞❡ ❛♥❡❧ ❡ ❝♦❛♥❡❧ sã♦ ❞❡✜♥✐❞❛s s♦❜r❡ ❞✐❢❡r❡♥t❡s á❧❣❡❜r❛s ❞❡ ❜❛s❡✳ ◆❛ ✈❡r❞❛❞❡✱ sã♦ ♠♦♥♦✐❞❡s ❡ ❝♦♠♦♥♦✐❞❡s ❡♠ ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s ❞✐st✐♥t❛s✱ ❛ s❛❜❡r✱ ❛ ❡str✉t✉r❛ ❞❡ ♦❜❥❡t♦ ❝♦á❧❣❡❜r❛ é ❞❡✜♥✐❞❛ s♦❜r❡ R ❡ k R op ❛ ❡str✉t✉r❛ ❞❡ ♦❜❥❡t♦ á❧❣❡❜r❛ é ❞❡✜♥✐❞❛ s♦❜r❡ R ⊗ ✳

  ❖ ♣ró①✐♠♦ ❧❡♠❛ q✉❡ ✈❛♠♦s ♣r♦✈❛r✱ ❝❛r❛❝t❡r✐③❛ R✲❛♥é✐s✳ P❡r❝❡❜❛ q✉❡✱ s❡ (A, µ, η) é ✉♠ R✲❛♥❡❧✱ ❡♥tã♦ A ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ❞❡ k✲ k A −→ A á❧❣❡❜r❛✳ ❆ ♠✉❧t✐♣❧✐❝❛çã♦ m : A ⊗ é ❞❡✜♥✐❞❛ ♣♦r m = µ ◦ π✱ k A −→ A ⊗ R A ❡♠ q✉❡ π : A ⊗ é ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛✱ ❡ ❛ ✉♥✐❞❛❞❡ η A : k −→ A A = η ◦ η R R

  é ❞❡✜♥✐❞❛ ♣♦r η ❡♠ q✉❡ η é ❛ ✉♥✐❞❛❞❡ ❞❛ k ✲á❧❣❡❜r❛ R✳ ❆ ❛çã♦ à ❞✐r❡✐t❛ ❡ ❛ ❛çã♦ à ❡sq✉❡r❞❛ ❞❡ k ❡♠ A sã♦ ❞❛❞❛s R (s) R (s)a r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r a · s = aη ❡ s · a = η ✱ ♣❛r❛ q✉❛✐sq✉❡r a ∈ A

  ❡ s ∈ k✳ ▲❡♠❛ ✷✳✶ ❊①✐st❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ✶ ❛ ✶ ❡♥tr❡ R✲❛♥é✐s (A, µ, η) ❡ ♠♦r✜s♠♦s ❞❡ k✲á❧❣❡❜r❛s η : R −→ A✳ ❉❡♠♦♥str❛çã♦✿ (=⇒) ❙❡❥❛ (A, µ, η) ✉♠ R✲❛♥❡❧✳ t❡♠♦s q✉❡ ♠♦str❛r q✉❡ η é ♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s✳ ❉❡ ❢❛t♦✱ ❝❧❛r♦ q✉❡ η é k✲❧✐♥❡❛r✳ ❇❛st❛ ✈❡r✐✜❝❛r♠♦s q✉❡ η é ♠✉❧t✐♣❧✐❝❛t✐✈♦

  η(rs) = η(r1 R s) = r · η(1 R ) · s = r · 1 A · s = (r · 1 A ) · s

  = ((r · 1 A )1 A ) · s = (r · 1 A )(1 A · s) = (r · η(1 R ))(η(1 R ) · s) = η(r)η(s).

  (⇐=) ❙✉♣♦♥❤❛ q✉❡ η : R −→ A s❡❥❛ ♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s✳ ❊♠

  ♣❛rt✐❝✉❧❛r✱ η é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣♦❞❡♠♦s ♠✉♥✐r A ❝♦♠ ❛ s❡❣✉✐♥t❡ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦✱ r · a · s = η(r)aη(s)✱ ♣❛r❛ q✉❛✐sq✉❡r r, s ∈ R ❡ a ∈ A✳ ❆ss✐♠✱

  η(rs) = η(r1 R s) = η(r)η(1 R )η(s) = r · 1 A · s. k A ❚❛♠❜é♠ ♣♦❞❡♠♦s ♠✉♥✐r A⊗ ❝♦♠ ❛ s❡❣✉✐♥t❡ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦✱ r · (a ⊗ b) · s = η(r)a ⊗ bη(s), k A

  ♣❛r❛ q✉❛✐sq✉❡r r, s ∈ R ❡ a ⊗ b ∈ A ⊗ ✳ ❉❡st❛ ❢♦r♠❛✱ ♥ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ m é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✳ P♦rt❛♥t♦✱ ♣r❡❝✐s❛♠♦s ❛♣❡♥❛s R A −→ A ❝♦♥str✉✐r ✉♠❛ ♠✉❧t✐♣❧✐❝❛çã♦ µ : A ⊗ ❛ q✉❛❧ é ♠♦r✜s♠♦ ❞❡ R

  ✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ❛♣❛rt✐r ❞❡ η ♣♦❞❡♠♦s ❝♦♥str✉✐r ❞♦✐s ♠♦r✜s♠♦s

  f, g : A ⊗ k R ⊗ k A −→ A ⊗ k

  A, ❞❡✜♥✐❞♦s ♣♦r f (a ⊗ r ⊗ b) = aη(r) ⊗ b k R ⊗ k A ❡ g(a ⊗ r ⊗ b) = a ⊗ η(r)b,

  ♣❛r❛ t♦❞♦ a ⊗ r ⊗ b ∈ A ⊗ ✳ ◆ã♦ é ❞✐❢í❝✐❧ ♠♦str❛r q✉❡ f, g sã♦ ♠♦r✜s♠♦s ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡st❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❜❛❧❛♥❝❡❛❞♦ ♣♦r R✱ ❝♦♠♦ ♦ ❝♦❡q✉❛❧✐③❛❞♦r f π

  // // A ⊗

  A ⊗ k R ⊗ k A k A R A. g // A ⊗ ❆❣♦r❛ ♣❡❧❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ m t❡♠♦s m ◦ f (a ⊗ r ⊗ b) = m(aη(r) ⊗ b)

  = (aη(r))b = a(η(r)b) = m(a ⊗ η(r)b) = m ◦ g(a ⊗ r ⊗ b). P♦rt❛♥t♦✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞♦ ❝♦❡q✉❛❧✐③❛❞♦r✱ ❡①✐st❡ ú♥✐❝♦ µ : A ⊗ R A −→ A

  ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✱ t❛❧ q✉❡✱ ♦ s❡❣✉✐♥t❡ ❞✐❛✲ ❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦✱ f π

  // //

  A ⊗ k R ⊗ k A k A A ⊗ R A g // A ⊗ m µ %% A. ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞❡ µ✳ k R op

  ❉❡ss❡ ❧❡♠❛✱ ✉♠ R ⊗ ✲❛♥❡❧ ❇ é ❞❡s❝r✐t♦ ♣♦r ✉♠ ♠♦r✜s♠♦ ❞❡ op k k R −→ B ✲á❧❣❡❜r❛s η : R ⊗ ✳ ❉❡ ♠❛♥❡✐r❛ ❡q✉✐✈❛❧❡♥t❡✱ ♣♦❞❡♠♦s

  ❝♦♥s✐❞❡r❛r ❛s r❡str✐çõ❡s op s := η(− ⊗ k

  1 R ) : R −→ B R ⊗ k −) : R −→ B, ❡ t := η(1 q✉❡ sã♦ ♠♦r✜s♠♦s ❞❡ k✲á❧❣❡❜r❛s ❝♦♠ ✐♠❛❣❡♥s ❝♦♠✉t❛♥❞♦ ❡♠ B✳ ❖s k R op

  ♠♦r✜s♠♦s s ❡ t sã♦ ❝❤❛♠❛❞♦s ❞❡ s♦✉r❝❡ ❡ t❛r❣❡t ❞❡ ✉♠ R ⊗ ✲ k R op ❛♥❡❧ ❇✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❯♠ R ⊗ ✲❛♥❡❧ s❡rá ❞❡♥♦t❛❞♦ ♣❡❧❛ tr✐♣❧❛ (B, s, t)

  ✱ ❡♠ q✉❡ B é ✉♠❛ k✲á❧❣❡❜r❛✱ ❡ s✱ t sã♦ ♠♦r✜s♠♦s ❞❡ á❧❣❡❜r❛s ❝♦♠ op k R ✐♠❛❣❡♥s ❝♦♠✉t❛♥❞♦ ❡♠ B✳ P♦r s✐♠♣❧✐❝✐❞❛❞❡ ✈❛♠♦s ❞❡♥♦t❛r R ⊗ e op ♣♦r R ❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ R ♣♦r r✳ r

  B R B ❆❣♦r❛ ❝♦♥s✐❞❡r❡ B × R ♦ k✲s✉❜♠ó❞✉❧♦ ❞❡ B ⊗ ❞❡✜♥✐❞♦ ♣♦r

  ( ) r

  X X

  X

  ′ ′ ′ B× B := b i ⊗b ∈ B ⊗ R B | s(r)b i ⊗b = b i ⊗t(r)b ∀ r ∈ R . R i i i i i i r

  B ➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ B × R é ✉♠❛ k✲á❧❣❡❜r❛✱ ❝♦♠ ♠✉❧t✐♣❧✐❝❛çã♦✱ r

  B ❞❛❞❛ ♣❛r❛ q✉❛✐sq✉❡r a ⊗ b, c ⊗ d ∈ B × R ✱ ♣♦r (a ⊗ b)(c ⊗ d) = ac ⊗ bd✳ ❉❡ ♠❛♥❡✐r❛ s✐♠étr✐❝❛ ❞❡✜♥✐♠♦s

  ( ) l

  X X

  X

  ′ ′ ′

  B× B := b i ⊗b ∈ B ⊗ R B | b i t(r) ⊗b = b i ⊗b s(r) ∀ r ∈ R , R i i i i i i q✉❡ t❛♠❜é♠ ✉♠❛ k✲á❧❣❡❜r❛✳

  ✷✳✶ ❉❡✜♥✐çã♦ ❡ ❊①❡♠♣❧♦s

  ❉❡✜♥✐çã♦ ✷✳✷ ❙❡❥❛ R ✉♠❛ á❧❣❡❜r❛ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k✳ ❯♠ e R

  ✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛ B✱ ❝♦♥s✐st❡ ❞❡ ✉♠ R ✲❛♥❡❧ (B, s, t) ❡ ❞❡ ✉♠ R✲❝♦❛♥❡❧ (B, ∆, ε) ♥♦ ♠❡s♠♦ k✲♠ó❞✉❧♦ B✱ q✉❡ ❡stã♦ s✉❥❡✐t♦s ❛♦s s❡❣✉✐♥t❡s ❛①✐♦♠❛s ❞❡ ❝♦♠♣❛t✐❜✐❞❛❞❡✿

  ✭✐✮ ❆ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦ ♥♦ R✲❝♦❛♥❡❧ (B, ∆, ε) é ❞❛❞❛ ♣♦r r · b · r

  X i b i ⊗t(r)b

  (2) P❛r❛ q✉❛✐sq✉❡r a, b ∈ B✱ t❡♠♦s

  ε(s(r)) = ε(1 B s(r)) = ε(1 B · r) = ε(1 B )r = r.

  = ε(r · 1 B ) = rε(1 B ) = r, t❛♠❜é♠ t❡♠♦s

  ❉❡♠♦♥str❛çã♦✿ (1) P❛r❛ t♦❞♦ r ∈ R✱ t❡♠♦s ε(t(r)) = ε(1 B t(r))

  (2) ε(t(ε(a))b) = ε(s(ε(a))b) ✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ B✳

  (1) ε(t(r)) = ε(s(r)) = r ✱ ♣❛r❛ t♦❞♦ r ∈ R❀

  ✲❛♥❡❧ ❞❛❞❛ ♣♦r (B, s, t) ❡ ❡str✉t✉r❛ ❞❡ R✲❝♦❛♥❡❧ ❞❛❞❛ ♣♦r (B, ∆, ε)✳ ❊♥tã♦ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿

  ♣r♦♣♦s✐çã♦✳ Pr♦♣♦s✐çã♦ ✷✳✸ ❙❡❥❛ B ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✱ ❝♦♠ ❡str✉t✉r❛ ❞❡ R e

  é ♦ R✲❛♥❡❧ ❞❡s❝r✐t♦ ♣♦r s ❝♦♠♦ ✉♥✐❞❛❞❡✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ✐t❡♠ (iii) ❞❛ ❉❡✜♥✐çã♦ ✷✳✷ t❡♠♦s ❛ s❡❣✉✐♥t❡

  ✭✐✐✐✮ ❆ ❝♦✉♥✐❞❛❞❡ ε é ✉♠ ❝❛r❛❝t❡r à ❞✐r❡✐t❛ ♥♦ R✲❛♥❡❧ (B, s)✳ ❊♠ q✉❡ (B, s)

  ∀ r ∈ R ) .

  ′ i

  =

  

  ′ i

  X i s(r)b i ⊗b

  ∈ B ⊗ R B |

  ′ i

  X i b i ⊗b

  B× r R B := (

  B ❡ ∆ é ♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s✱ ❡♠ q✉❡

  ∈ R ❡ b ∈ B. ✭✐✐✮ ❈♦♥s✐❞❡r❛♥❞♦ ❛ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦ ❡♠ B✱ ❝♦♠♦ ❡♠ ✷✳✶✱ t❡♠♦s ∆(B) ⊆ B × R r

  ′

  ♣❛r❛ q✉❛✐sq✉❡r r, r

  )t(r) ✭✷✳✶✮

  ′

  = bs(r

  ε(t(ε(a))b) = ε(s(ε(t(ε(a))))b)

  = ε(s(ε(a))b). r B R B

  ◆♦t❡ q✉❡ ♦ k✲s✉❜♠ó❞✉❧♦ B× R ❞❡ B⊗ é ❞❡✜♥✐❞♦ ❞❡ t❛❧ ♠❛♥❡✐r❛ r B q✉❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ é ❜❡♠ ❞❡✜♥✐❞❛✳ B × R é ❝❤❛♠❛❞♦ ❞❡ ♣r♦❞✉t♦ r

  B ❚❛❦❡✉❝❤✐ à ❞✐r❡✐t❛✳ ◆❛ ✈❡r❞❛❞❡ B × R t❡♠ ♠❛✐s ❡str✉t✉r❛ ❞♦ q✉❡ ✉♠❛ k✲á❧❣❡❜r❛✱ ❝♦♠♦ ✈❛♠♦s ✈❡r ♥❛ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦✳ ❆♥t❡s✱ ❡♥✉♥❝✐❛r❡♠♦s ✉♠ r❡s✉❧t❛❞♦ q✉❡ ✈❛♠♦s ✉s❛r✳ ▲❡♠❛ ✷✳✹ ✭❊❧❡♠❡♥t♦ ♥✉❧♦ ❡♠ ✉♠ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧✮ ❙❡❥❛ R ✉♠ ❛♥❡❧ R R i } i M ❝♦♠ ✉♥✐❞❛❞❡✱ N ∈ ❡ M ∈ M ✳ ❙❡❥❛♠ ❛✐♥❞❛ {n ∈I ✉♠❛ ❢❛♠í❧✐❛ i } i ❞❡ ❣❡r❛❞♦r❡s ♣❛r❛ N ❡ {m ∈I ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ M t❛✐s i = 0 q✉❡ m ❛ ♠❡♥♦s ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ í♥❞✐❝❡s✳ ❊♥tã♦ ♣❛r❛

  X i m i ⊗ n i ∈ M ⊗ R N

  ∈I

  P m i ⊗ n i = 0 t❡♠♦s q✉❡ i s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠❛ ❢❛♠í❧✐❛ ✜♥✐t❛

  ∈I

  {x j } j ∈J ji } ❡♠ M ❡ ✉♠❛ ❢❛♠í❧✐❛ {r (j,i)∈J×I ❡♠ R s❛t✐s❢❛③❡♥❞♦

  (i) r ji 6= 0 ❛♣❡♥❛s ♣❛r❛ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ í♥❞✐❝❡s (j, i) ∈ J ×I❀

  P (ii) r ji · n i = 0 i ♣❛r❛ t♦❞♦ j ∈ J❀

  ∈I

  P (iii) x j · r ji = m i j ✳

  ∈J

  ❊st❡ r❡s✉❧t❛❞♦ ❡♥❝♦♥tr❛✲s❡ ♣r♦✈❛❞♦ ❡♠ ❬✸✵❪ ♣á❣✐♥❛ ✾✼✳ e Pr♦♣♦s✐çã♦ ✷✳✺ ❙❡❥❛ (B, s, t) ✉♠ R ✲❛♥❡❧✳ ❊♥tã♦ ♦ ♣r♦❞✉t♦ ❚❛❦❡✉❝❤✐ r e op r B × B k R −→ B × B R é ✉♠ R ✲❛♥❡❧ ❝♦♠ ✉♥✐❞❛❞❡ η : R ⊗ R ❞❛❞❛ ♣♦r

  ′ ′

  η(r ⊗ r ) = t(r ) ⊗ R s(r) ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛❞❛ ♣♦r

  X X

  X ( a i ⊗ R b i )( c j ⊗ R d j ) = a i c j ⊗ R b i d j , i j i,j

  P P r ′ op a i ⊗ R b i , c j ⊗ R d j ∈ B × B ∈ R ⊗ k R ♣❛r❛ q✉❛✐sq✉❡r R ❡ r ⊗ r ✳ i j ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r♦ ✈❛♠♦s ✈❡r q✉❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♦✉ s❡❥❛✱ q✉❡ ✐♥❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❡ r❡♣r❡s❡♥t❛♥t❡s ❞❡ ✉♠❛ ❝❧❛ss❡✳ ❇❛st❛ ♠♦str❛r♠♦s q✉❡ ♦ ♣r♦❞✉t♦ é ③❡r♦ q✉❛♥❞♦ ✉♠ ❞♦s ❢❛t♦r❡s

  P P P r a i ⊗ b i c j ⊗ d j ∈ B × B c j ⊗ R d j = 0 é ③❡r♦✳ ❉❡ ❢❛t♦✱ ♣❛r❛ ✱ R ✱ s❡ i j j kj ∈ R ♣❡❧♦ ❧❡♠❛ ✷✳✹✱ ❡①✐st❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ❡❧❡♠❡♥t♦s r ❡ x k ∈ B t❛✐s q✉❡

  X X

  X X r kj · d j = d j t(r kj ) = 0 x k · r kj = x k s(r kj ) = c j , j ik k

  ❡♥tã♦ t❡♠♦s

  X k b i t(r ki ) = 0 ❡

  ❝❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ q✉❡ 1 B ⊗ 1 B é ❛ ✉♥✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡st❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛✳ ❆❣♦r❛ ✈❛♠♦s ✈❡r✐✜❝❛r q✉❡ η é ✉♠ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s✳ ❉❡ ❢❛t♦✱ s❡❥❛♠ r, w, u, v ∈ R✱ ❡♥tã♦

  P j c j ⊗ d j ∈ B × R r B ✳ P♦rt❛♥t♦✱ ❛ ♠✉❧t✐♣❧✐✲

  X i,j,k y k c j ⊗ b i t(r ki )d j = 0. ❆q✉✐ ✉s❛♠♦s ♦ ❢❛t♦ q✉❡

  X i,j,k y k s(r ki )c j ⊗ b i d j =

  X i,j a i c j ⊗ b i d j =

  X j c j ⊗ d j   =

  

  X i a i ⊗ b i ! 

  X k y k s(r ki ) = a i , ❡♥tã♦ t❡♠♦s

  ✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ❡❧❡♠❡♥t♦s r ki ∈ R ❡ y k ∈ B t❛✐s q✉❡

  X i a i ⊗ b i ! 

  ♥❡❝❡ssár✐♦ ❛♣❡♥❛s ♥♦ ❝❛s♦ q✉❛♥❞♦ ♦ ♣r✐♠❡✐r♦ ❢❛t♦r é ③❡r♦✳ ❉❡ ❢❛t♦✱ s❡❥❛ P i a i ⊗ b i = 0

  P i a i ⊗ R b i ∈ B × R r B ✱ ✐ss♦ ✈❛✐ s❡r

  X i,j,k a i x k ⊗ b i d j t(r kj ) = 0. P❡r❝❡❜❛ q✉❡ ♥ã♦ ✉s❛♠♦s ♦ ❢❛t♦ q✉❡

  X i,j,k a i x k ⊗ r kj · b i d j =

  X i,j,k a i x k · r kj ⊗ b i d j =

  X i,j,k a i x k s(r kj ) ⊗ b i d j =

  X i,j a i c j ⊗ b i d j =

  X j c j ⊗ d j   =

  

  η((r ⊗ w)(u ⊗ v)) = η(ru ⊗ vw) = t(vw) ⊗ R s(ru)

  = t(w)t(v) ⊗ R s(r)s(u) = (t(w) ⊗ R s(r))(t(v) ⊗ R s(u)) R B = η(r ⊗ w)η(u ⊗ v). e

  ❙❡❣✉❡ ❡♥tã♦ q✉❡ B × é ✉♠ R ✲❛♥❡❧✳ ❉❡✜♥✐çã♦ ✷✳✻ ❙❡❥❛ L ✉♠❛ á❧❣❡❜r❛ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k✳ ❯♠ e L

  ✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛ B✱ ❝♦♥s✐st❡ ❞❡ ✉♠ L ✲❛♥❡❧ (B, s, t) ❡ ❞❡ ✉♠ L✲❝♦❛♥❡❧ (B, ∆, ε) ♥♦ ♠❡s♠♦ k✲♠ó❞✉❧♦ B✱ q✉❡ ❡stã♦ s✉❥❡✐t♦s ❛♦s s❡❣✉✐♥t❡s ❛①✐♦♠❛s ❞❡ ❝♦♠♣❛t✐❜✐❞❛❞❡✿

  ✭✐✮ ❆ ❡str✉t✉r❛ ❞❡ L✲❜✐♠ó❞✉❧♦ ♥♦ L✲❝♦❛♥❡❧ (B, ∆, ε) é ❞❛❞❛ ♣♦r

  ′ ′

  l · b · l = s(l)t(l )b ✭✷✳✷✮

  ′

  ∈ L ♣❛r❛ q✉❛✐sq✉❡r l, l ❡ b ∈ B. ✭✐✐✮ ❈♦♥s✐❞❡r❛♥❞♦ ❛ ❡str✉t✉r❛ ❞❡ L✲❜✐♠ó❞✉❧♦ ❡♠ B✱ ❝♦♠♦ ❡♠ ✷✳✷✱ l

  B t❡♠♦s ∆(B) ⊆ B × ❡ ∆ é ♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s✱ ❡♠ q✉❡ L (

  ) l

  X X

  X

  ′ ′ ′ B× B := b i ⊗b ∈ B ⊗ L B | b i t(l) ⊗b = b i ⊗b s(l) ∀ l ∈ L . R i i i i i i

  ✭✐✐✐✮ ❆ ❝♦✉♥✐❞❛❞❡ ε é ✉♠ ❝❛r❛❝t❡r à ❡sq✉❡r❞❛ ♥♦ L✲❛♥❡❧ (B, s)✳ ❊♠ q✉❡ (B, s) é ♦ L✲❛♥❡❧ ❞❡s❝r✐t♦ ♣♦r s ❝♦♠♦ ✉♥✐❞❛❞❡✳ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❛ Pr♦♣♦s✐çã♦ ✷✳✸✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ B ❡ l ∈ L

  ✱ t❡♠♦s ε(t(l)) = ε(s(l)) = l

  ✭✷✳✸✮ ❡ ε(at(ε(a))) = ε(as(ε(a))).

  ✭✷✳✹✮ ❆♥t❡s ❞❡ ❝♦♥t✐♥✉❛r♠♦s✱ ♥♦t❡ q✉❡ ❡♠ ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛ B✱ ♣❛r❛ q✉❛✐sq✉❡r b ∈ B ❡ r ∈ R✱ t❡♠♦s

  ∆(b · r) = ∆(b) · r = b ⊗ R b · r = b ⊗ R b s(r)

  (1) (2) (1) (2) ✭✷✳✺✮

  ❡ ∆(r · b) = r · ∆(b) = r · b ⊗ R b = b t(r) ⊗ R b .

  (1) (2) (1) (2) ✭✷✳✻✮

  ❏á ❡♠ ✉♠ L✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛ B✱ ♣❛r❛ q✉❛✐sq✉❡r b ∈ B ❡ l ∈ L✱ t❡♠♦s ∆(b · l) = ∆(b) · l = b ⊗ L b · l = b ⊗ L t(l)b

  (1) (2) (1) (2) ✭✷✳✼✮

  ❡ ∆(l · b) = l · ∆(b) = l · b ⊗ L b = s(l)b ⊗ L b .

  (1) (2) (1) (2) ✭✷✳✽✮

  ❉❡✜♥✐çã♦ ✷✳✼ ❙❡❥❛♠ L ❡ eL ❞✉❛s á❧❣❡❜r❛s s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k L ✱ B ❡ B ✱ L✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛ ❡ eL✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛✱ r❡s✲ L e e L

  ♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡♥♦t❡ ❛s ❡str✉t✉r❛s ❞❡ L ✲❛♥❡❧ ❡ L✲❝♦❛♥❡❧ ❡♠ B ✱ ♣♦r (B L , s L , t L ) L , ∆ L , ε L )

  ❡ (B ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡♥♦t❡ t❛♠❜é♠✱ ❛s ❡s✲ e , s , t ) , ∆ , ε ) tr✉t✉r❛s ❞❡ eL ✲❛♥❡❧ ❡ eL✲❝♦❛♥❡❧ ❡♠ B L e ✱ ♣♦r (B L e L e L e ❡ (B L e L e L e ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❯♠ ♠♦r✜s♠♦ ❞❡ ❜✐❛❧❣❡❜ró✐❞❡s à ❡sq✉❡r❞❛✱ é L −→ B , ϕ : L −→ e L) ✉♠ ♣❛r ❞❡ ♠♦r✜s♠♦s ❞❡ k✲á❧❣❡❜r❛s (Φ : B L e s❛t✐s❢❛③❡♥❞♦ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿

  (i) s ◦ ϕ = Φ ◦ s L L e ❀ (ii) t ◦ ϕ = Φ ◦ t L L e ❀

  (iii) ∆ ◦ Φ = (Φ ⊗ L Φ) ◦ ∆ L L e ❀ (iv) ε ◦ Φ = ϕ ◦ ε L L e ✳

  ❆♥❛❧♦❣❛♠❡♥t❡✱ ❞❡✜♥❡✲s❡ ♠♦r✜s♠♦ ❞❡ ❜✐❛❧❣❡❜ró✐❞❡s à ❞✐r❡✐t❛✳ Pr♦♣♦s✐çã♦ ✷✳✽ ❙❡❥❛ B ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❊♥tã♦ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿ op

  (1) B cop op é ✉♠ R ✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛❀ (2) B

  é ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛✳ cop ⊗R = t, t = s) op ′ ′ ❉❡♠♦♥str❛çã♦✿ (1) ◆♦t❡ q✉❡ B é ✉♠ R ✲❛♥❡❧ (B, s op cop , ∆ cop , ε) ❡ ♣♦ss✉✐ ❛ s❡❣✉✐♥t❡ ❡str✉t✉r❛ ❞❡ R ✲❝♦❛♥❡❧ (B ✳ ❆ ❡str✉t✉r❛ op cop , ∆ cop , ε) ∈ R

  ❞❡ R ✲❜✐♠ó❞✉❧♦ ❡♠ (B é ❞❛❞❛✱ ♣❛r❛ q✉❛✐sq✉❡r r, r ❡ b ∈ B ✱ ♣♦r

  ′ ′ ′ ′ ′ ′

  r ⊲ b ⊳ r = bt (r)s (r ) = bs(r)t(r ) = r · b · r. cop op ❉❡ss❛ ❢♦r♠❛✱ ∆ ❡ ε sã♦ ❝❧❛r❛♠❡♥t❡ ♠♦r✜s♠♦s ❞❡ R ✲❜✐♠ó❞✉❧♦s✳ cop ⊆ B cop × op B cop r ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ∆ R ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r b ∈ B

  ❡ r ∈ R✱ t❡♠♦s op op b ⊳ r ⊗ R b = b t(r) ⊗ R b

  

(2) (1) (2) (1)

  = r · b

  = b

  (1)

  ⊗ R op b

  (2)

  ❚❛♠❜é♠ t❡♠♦s (B cop ⊗ R op ε)∆ cop (b) = (B cop ⊗ R op ε)(b

  (1) .

  ⊗ R op b

  (2)

  ⊗ R op b

  

(3)

  (1)(1)

  

(2)

  ⊗ R op b

  (1)(2)

  ⊗ R op b

  

(2)

  ) = b

  (1)

  ⊗ R op b

  (2)

  (B cop ⊗ R op ∆ cop )∆ cop (b) = (B cop ⊗ R op ∆ cop )(b

  , ♣♦r ♦✉tr♦ ❧❛❞♦✱

  ) = b

  ⊳ ε(b

  ⊗ R op b

  ) ⊲ b

  ′

  ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ B✱ t❡♠♦s ε(s

  = t) ✳ ❉❡

  ′

  ) = b. ▼♦str❡♠♦s q✉❡ ε é ❝❛r❛❝t❡r à ❞✐r❡✐t❛ ♥♦ R op ✲❛♥❡❧ (B cop , s

  (2)

  · ε(b

  

(1)

  = b

  (1)

  (2)

  (1)

  ) = ε(b

  (1)

  ⊗ R op b

  (2)

  (ε ⊗ R op B cop )∆ cop (b) = (ε ⊗ R op B cop )(b

  = b, ❡

  (2)

  ) · b

  (1)

  ) = ε(b

  (1)

  (2)

  (2)

  (2)

  = t(r)b

  (1)

  ⊗ R op b

  (2)

  (r)b

  ′

  , t❛♠❜é♠ t❡♠♦s s

  (1)

  ⊗ R op r ⊲ b

  s(r) = b

  ⊗ R op b

  (1)

  ⊗ R op b

  (2)

  · r = b

  (1)

  ⊗ R op b

  (2)

  = b

  (1)

  ⊗ R op b

  (2)

  (1)

  ⊗ R op b

  (2)

  

(3)

  = b

  (1)

  ⊗ R op b

  (2)(1)

  ⊗ R op b

  

(2)(2)

  ) = b

  (1)

  ⊗ R op b

  (∆ cop ⊗ R op B cop )∆ cop (b) = (∆ cop ⊗ R op B cop )(b

  = b

  ▼♦str❡♠♦s q✉❡ ✈❛❧❡♠ ❛ ❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ ❛ ❝♦✉♥✐t❛❧✐❞❛❞❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ b ∈ B✱ t❡♠♦s

  (1) .

  (r)b

  ′

  ⊗ R op t

  (2)

  = b

  (1)

  ⊗ R op s(r)b

  (2)

  (ε(a))b) = ε(t(ε(a))b)

  = ε(s(ε(a))b) cop op = ε(ab). ❙❡❣✉❡ ♣♦rt❛♥t♦✱ q✉❡ B é ✉♠ R ✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ op e

  ′ ′

  (2) = t, t = s) ◆♦t❡ q✉❡ B é ✉♠ R ✲❛♥❡❧ (B, s ❡ ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ op op

  , ∆, ε) ❞❡ R✲❝♦❛♥❡❧ (B ✳ ❆ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦ ❡♠ B é ❞❛❞❛✱

  ′

  ∈ R ♣❛r❛ q✉❛✐sq✉❡r r, r ❡ b ∈ B✱ ♣♦r

  

′ ′ ′ ′

  r ⊲ b ⊳ r = s (r) · op t (r ) · op b

  ′

  = t(r) · op s(r ) · op b

  ′

  = bt(r)s(r ), op , ∆, ε)

  ♦✉ s❡❥❛✱ ❛ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦ ❡♠ (B é ❛ ♠❡s♠❛ ❡♠ op op l op (B, ∆, ε) ) ⊆ B × B op ✳ ▼♦str❡♠♦s q✉❡ ∆(B ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ R b ∈ B

  ✱ t❡♠♦s

  ′

  b · op t (r) ⊗ R b = s(r)b ⊗ R b

  (1) (2) (1) (2)

  = b ⊗ R t(r)b

  (1) (2)

  = b ⊗ R b · op t(r)

  (1) (2) ′

  = b ⊗ R b · op s (r).

  (1) (2) op

  , s = t) ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ε é ❝❛r❛❝t❡r à ❡sq✉❡r❞❛ ❡♠ (B ✳ ❉❡ ❢❛t♦✱ op ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ B ✱ t❡♠♦s

  ε(b · op t(ε(a))) = ε(t(ε(a))b) = ε(s(ε(a))) = ε(ab) op = ε(b · op a).

  ❈♦♥❝❧✉í♠♦s ♣♦rt❛♥t♦✱ q✉❡ B é ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛✳ ❊①❡♠♣❧♦ ✷✳✾ ❇✐á❧❣❡❜r❛s A s♦❜r❡ k✱ sã♦ k✲❜✐❛❧❣❡❜ró✐❞❡s ♥❛ ❝❛t❡❣♦r✐❛ ❞♦s k✲♠ó❞✉❧♦s ✭à ❡sq✉❡r❞❛ ♦✉ à ❞✐r❡✐t❛✮✳ ❖s ♠♦r✜s♠♦s s ❡ t sã♦ ✐❣✉❛✐s ❛♦ ♠♦r✜s♠♦ ✉♥✐❞❛❞❡ η : k −→ A✳ k k k k op op ❉❡ ❢❛t♦✱ ♥♦t❡ q✉❡ k ≃ k ⊗ ✱ ♣♦rt❛♥t♦ A é k ⊗ ✲❛♥❡❧✱ ♣♦✐s é k✲ á❧❣❡❜r❛ ❡ η é ♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s✳ ❱❡r✐✜❝❛♥❞♦ ♦s ✐t❡♥s ❞❛ ❞❡✜♥✐çã♦ r P

  ′

  A = A ⊗ k A b i ⊗ k b ∈ A ⊗ k A t❡♠♦s✱ A × k ✱ ❞❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r i ❡ r ∈ k✱ t❡♠♦s

  X X

  ′ ′ i i η(r)b i ⊗ b i = r · b i ⊗ b i X

  ′

  = b i · r ⊗ b i i

  X

  ′

  = b i ⊗ r · b i i

  X

  ′ = b i ⊗ η(r)b . i i A ) = 1 k

  ❱❛♠♦s ✈❡r✐✜❝❛r q✉❡ ε é ❝❛r❛❝t❡r à ❞✐r❡✐t❛ ❞❡ (A, η)✳ ❉❡ ❢❛t♦✱ ε(1 é ❝❧❛r♦✱ ♣♦✐s ε é ♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛✳ P❛r❛ q✉❛✐sq✉❡r a, b ∈ A ❡ r ∈ k✱ t❡♠♦s

  ε(aη(r)) = ε(a · r) = ε(a)r ❡ ε(η(ε(a))b) = ε(ε(a) · b) = ε(a)ε(b) = ε(ab).

  P♦rt❛♥t♦✱ t❡♠♦s q✉❡ A é ✉♠ k✲❜✐❛❧❣❡❜r♦✐❞❡ à ❞✐r❡✐t❛ ✭à ❡sq✉❡r❞❛✮✳ ❊①❡♠♣❧♦ ✷✳✶✵ ❙❡❥❛ R ✉♠❛ á❧❣❡❜r❛ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k✳ P♦✲ e ❞❡♠♦s ♠✉♥✐r R ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❖s ♠♦r✜s♠♦s s♦✉r❝❡ ❡ t❛r❣❡t sã♦ ❞❛❞♦s ♣❡❧❛s ✐♥❝❧✉sõ❡s e op e s : R −→ R −→ R ,

  ❡ t : R

R 7−→ 1 R ⊗ r

′ ′ ❞❡✜♥✐❞♦s ♣♦r r 7−→ r ⊗ 1 ❡ r ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣❛r❛

  ′

  ∈ R q✉❛✐sq✉❡r r, r ✳ ❖ ❝♦♣r♦❞✉t♦ é ❞❡✜♥✐❞♦ ♣♦r e e e ∆ : R −→ R ⊗ R R

  ′ ′

  r ⊗ r 7−→ (1 R ⊗ r ) ⊗ R (r ⊗ 1 R ), ❡ ❛ ❝♦✉♥✐❞❛❞❡ é ❞❡✜♥✐❞❛ ♣♦r e

  ε : R −→ R

  ′ ′ op r ⊗ r 7−→ r r, ′

  ∈ R ⊗ k R ♣❛r❛ t♦❞♦ r ⊗ r ✳ e e Pr✐♠❡✐r♦✱ é ❝❧❛r♦ q✉❡ R é ✉♠ R ✲❛♥❡❧✱ ❝♦♠ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛❞❛ ❡♥✲ tr❛❞❛ à ❡♥tr❛❞❛ ❡ ✉♥✐❞❛❞❡ ❞❛❞❛ ♣❡❧❛ ✐❞❡♥t✐❞❛❞❡✳ ❱❛♠♦s ✈❡r ❡♥tã♦ q✉❡ e

  (R , ∆, ε) é ✉♠ R✲❝♦❛♥❡❧✳ ❱❡r❡♠♦s ♣r✐♠❡✐r♦ q✉❡ ∆ ❡ ε sã♦ ♠♦r✜s♠♦s e

  ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R ❡ a ⊗ b ∈ R ✱ t❡♠♦s ∆(r · (a ⊗ b)) = ∆((a ⊗ b)t(r))

  = ∆((a ⊗ b)(1 R ⊗ r)) = ∆(a ⊗ br)

  = (1 R ⊗ br) ⊗ R (a ⊗ 1 R ) = (1 R ⊗ b)(1 R ⊗ r) ⊗ R (a ⊗ 1 R ) = (1 R ⊗ b)t(r) ⊗ R (a ⊗ 1 R ) = r · (1 R ⊗ b) ⊗ R (a ⊗ 1 R ) = r · ∆(a ⊗ b)

  ❡ ∆((a ⊗ b) · r) = ∆((a ⊗ b)(r ⊗ 1 R ))

  = ∆(ar ⊗ b) = (1 R ⊗ b) ⊗ R (ar ⊗ 1 R ) = (1 R ⊗ b) ⊗ R (a ⊗ 1 R )(r ⊗ 1 R ) = (1 R ⊗ b) ⊗ R (a ⊗ 1 R )s(r) = (1 R ⊗ b) ⊗ R (a ⊗ 1 R ) · r = ∆(a ⊗ b) · r.

  ❚❛♠❜é♠ t❡♠♦s ε(r · (a ⊗ b)) = ε((a ⊗ b)(1 R ⊗ r))

  = ε(a ⊗ br) = ε(a ⊗ rb) = (rb)a = r(ba) = rε(a ⊗ b)

  ❡ ε((a ⊗ b) · r) = ε((a ⊗ b)(r ⊗ 1 R ))

  = ε(ar ⊗ b) = b(ar) = (ba)r = ε(a ⊗ b)r.

  ❱❛♠♦s ✈❡r ❛❣♦r❛ q✉❡ e e e e e (R ⊗ ∆) ◦ ∆ = (∆ ⊗ R ) ◦ ∆ ) ◦ ∆ = I R = (R ⊗ ε) ◦ ∆. e ❡ (ε ⊗ R ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ a ⊗ b ∈ R ✱ t❡♠♦s e e

  (R ⊗ ∆) ◦ ∆(a ⊗ ¯b) = (R ⊗ ∆)[(1 R ⊗ ¯b) ⊗ R (a ⊗ 1 R )]

  = (1 R ⊗ ¯b) ⊗ R [(1 R ⊗ 1 R ) ⊗ R (a ⊗ 1 R )] = [(1 R ⊗ ¯b) ⊗ R (1 R ⊗ 1 R )] ⊗ R (a ⊗ 1 R ) = (∆ ⊗ R e )[(1 R ⊗ ¯b) ⊗ R (a ⊗ 1 R )] = (∆ ⊗ R e ) ◦ ∆(a ⊗ ¯b)

  ❡ t❛♠❜é♠✱ (ε ⊗ R e ) ◦ ∆(a ⊗ ¯b) = (ε ⊗ R e )[(1 R ⊗ ¯b) ⊗ R (a ⊗ 1 R )]

  = ε(1 R ⊗ ¯b) · (a ⊗ 1 R ) = b · (a ⊗ 1 R ) = (a ⊗ 1 R )t(b) = (a ⊗ 1 R )(1 R ⊗ b) = (a ⊗ b),

  (R e ⊗ ε) ◦ ∆(a ⊗ ¯b) = (R e ⊗ ε)[(1 R ⊗ ¯b) ⊗ R (a ⊗ 1 R )] = (1 R ⊗ ¯b) · ε(a ⊗ 1 R ) = (1 R ⊗ ¯b) · a = (1 R ⊗ ¯b)s(a) = (1 R ⊗ ¯b)(a ⊗ 1 R ) = (a ⊗ ¯b).

  P♦rt❛♥t♦✱ t❡♠♦s q✉❡ (R e , ∆, ε)

  é ✉♠ R✲❝♦❛♥❡❧✳ ❱❛♠♦s ✈❡r✐✜❝❛r q✉❡ ❛ ✐♠❛❣❡♠ ❞❡ ∆ ❡stá ❝♦♥t✐❞❛ ♥♦ ♣r♦❞✉t♦ ❚❛❦❡✉❝❤✐ à ❞✐r❡✐t❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R ❡ a ⊗ b ∈ R e ✱ t❡♠♦s s(r)(1 R ⊗ ¯b) ⊗ R (a ⊗ 1 R ) = (r ⊗ 1 R )(1 R ⊗ ¯b) ⊗ R (a ⊗ 1 R )

  = (r ⊗ ¯b) ⊗ R (a ⊗ 1 R ) = (1 R ⊗ ¯b)(r ⊗ 1 R ) ⊗ R (a ⊗ 1 R ) = (1 R ⊗ ¯b)s(r) ⊗ R (a ⊗ 1 R ) = (1 R ⊗ ¯b) · r ⊗ R (a ⊗ 1 R ) = (1 R ⊗ ¯b) ⊗ R r · (a ⊗ 1 R ) = (1 R ⊗ ¯b) ⊗ R (a ⊗ 1 R )t(r) = (1 R ⊗ ¯b) ⊗ R (a ⊗ 1 R )(1 R ⊗ ¯ r) = (1 R ⊗ ¯b) ⊗ R (a ⊗ ¯ r) = (1 R ⊗ ¯b) ⊗ R (1 R ⊗ ¯ r)(a ⊗ 1 R ) = (1 R ⊗ ¯b) ⊗ R t(r)(a ⊗ 1 R ).

  ❙❡❣✉❡ ♣♦rt❛♥t♦✱ q✉❡ ❛ ✐♠❛❣❡♠ ❞❡ ∆ ❡stá ❝♦♥t✐❞❛ ♥♦ ♣r♦❞✉t♦ ❚❛❦❡✉❝❤✐ à ❞✐r❡✐t❛✳ ❆❣♦r❛ ✈❛♠♦s ✈❡r q✉❡ ∆ é ♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s✳ ❇❛st❛ ✈❡r✐✜❝❛r♠♦s q✉❡ ∆ é ♠✉❧t❧✐❝❛t✐✈♦✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a⊗b, c⊗d ∈ e R

  ✱ t❡♠♦s R R R R R ∆(a ⊗ ¯b)∆(c ⊗ ¯

  d) = [(1 ⊗ ¯b) ⊗ (a ⊗ 1 )][(1 ⊗ ¯

  d) ⊗ (c ⊗ 1 )] = (1 R ⊗ ¯b)(1 R ⊗ ¯

  d) ⊗ R (a ⊗ 1 R )(c ⊗ 1 R ) = (1 R ⊗ ¯b ¯

  d) ⊗ R (ac ⊗ 1 R ) = ∆(ac ⊗ ¯b ¯

  d) = ∆((a ⊗ ¯b)(c ⊗ ¯ d).

  P♦rt❛♥t♦✱ ∆ é ♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s✳ ❘❡st❛ ✈❡r✐✜❝❛r♠♦s q✉❡ ε é ✉♠ e , s)

  ❝❛r❛❝t❡r à ❞✐r❡✐t❛ ♥♦ R✲❛♥❡❧ (R ✳ ❉❡ ❢❛t♦✱ t❡♠♦s ε(1 R ⊗ 1 R ) = 1 R

  1 R = 1 R , t❛♠❜é♠ t❡♠♦s ε(s(ε(a ⊗ ¯b))(c ⊗ ¯

  d)) = ε((ba ⊗ 1 R )(c ⊗ ¯

  d)) = ε(bac ⊗ ¯

  d) = d(bac) = (db)(ac) = ε(ac ⊗ db) = ε(ac ⊗ ¯b ¯

  d) e = ε((a ⊗ ¯b)(c ⊗ ¯

  d)), ♣❛r❛ q✉❛✐sq✉❡r a⊗b, c⊗d ∈ R ✳ ❙❡❣✉❡ ❞❛í q✉❡ ε é ✉♠ ❝❛r❛❝t❡r à ❞✐r❡✐t❛ e e

  , s) ♥♦ R✲❛♥❡❧ (R ✳ P♦rt❛♥t♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ R é ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❊①❡♠♣❧♦ ✷✳✶✶ ✭❇✐á❧❣❡❜r❛s ❢r❛❝❛s✮ ❯♠❛ ❜✐á❧❣❡❜r❛ ❢r❛❝❛ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k✱ ❝♦♥s✐st❡ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❡ ✉♠❛ ❝♦á❧❣❡❜r❛ s♦❜r❡ ♦ ♠❡s♠♦ k✲♠ó❞✉❧♦ B✱ q✉❡ ❡stã♦ s✉❥❡✐t❛s ❛ ❛①✐♦♠❛s ❞❡ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ q✉❡ ❣❡♥❡r❛❧✐③❛♠ ♦s ❛①✐♦♠❛s ❞❡ ❜✐á❧❣❡❜r❛s✳ ❉❡ ♠❛♥❡✐r❛ ♠❛✐s ❝❧❛r❛✱ ♦ ❝♦✲ ♣r♦❞✉t♦ ❝♦♥t✐♥✉❛ s❡♥❞♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✱ ♠❛s ❛ ✉♥✐t❛❧✐❞❛❞❡ ❞♦ ❝♦✲♣r♦❞✉t♦ ∆

  ❡ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈✐❞❛❞❡ ❞❛ ❝♦✉♥✐❞❛❞❡ ε sã♦ ❡♥❢r❛q✉❡❝✐❞❛s✱ ❝♦♥❢♦r♠❡ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s

  (∆(1 B ) ⊗ k

  1 B )(1 B ⊗ k ∆(1 B )) = (∆ ⊗ k B)∆(1 B ) = (1 B ⊗ k ∆(1 B ))(∆(1 B ) ⊗ k

  1 B ) ✭✷✳✾✮

  ❡ ε(ab )ε(b

  c) = ε(abc) = ε(ab )ε(b

  c),

  (1) (2) (2) (1) ✭✷✳✶✵✮

  ♣❛r❛ q✉❛✐sq✉❡r a, b, c ∈ B✳ ◆♦t❡ q✉❡ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ❝♦♥❞✐çã♦ ✷✳✾ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛ ′ ′ ′ ′ 1 ⊗ k

  1 1 ⊗ k 1 = 1 ⊗ k 1 ⊗ k 1 = 1 ⊗ k

  1 1 ⊗ k

  1

  

(1) (1 ) (2) (2 ) (1) (2) (3) (1) (2) (1 ) (2 )

  ❈♦♥s✐❞❡r❡✱ ♣❛r❛ t♦❞♦ b ∈ B✱ ❛ s❡❣✉✐♥t❡ ❛♣❧✐❝❛çã♦ ✐❞❡♠♣♦t❡♥t❡ R ⊓ : B −→ B, b 7−→ 1 ε(b1 ). R (1) (2) )

  ❚❡♠♦s q✉❡ R := Im(⊓ é ✉♠❛ s✉❜á❧❣❡❜r❛ ❞❡ B✳ ❚❡♠♦s ❛✐♥❞❛ q✉❡ e B

  é ✉♠ R ✲❛♥❡❧✱ ❝♦♠ s♦✉r❝❡ s : R −→ B ❞❛❞♦ ♣❡❧❛ ✐♥❝❧✉sã♦ ❡ t❛r❣❡t t : R −→ B ❞❛❞♦✱ ♣❛r❛ t♦❞♦ b ∈ R✱ ♣❡❧❛ r❡str✐çã♦ ❞❛ ❛♣❧✐❝❛çã♦ t : B −→ B, b 7−→ ε(b1 )1 .

  (1) (2)

  P♦❞❡♠♦s ♠✉♥✐r B ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ R✲❝♦❛♥❡❧✳ ❖ ❝♦✲♣r♦❞✉t♦ é k B −→ B ⊗ R B ❞❡✜♥✐❞♦ ♣♦r ∆ := π ◦ ∆✱ ❡♠ q✉❡ π : B ⊗ ❡ ❛ ❝♦✉♥✐❞❛❞❡ R é ❞❡✜♥✐❞❛ ♣♦r ε := ⊓ ✳ e

  Pr✐♠❡✐r❛♠❡♥t❡ ✈❛♠♦s ✈❡r q✉❡ (B, s, t) é ❞❡ ❢❛t♦✱ ✉♠ R ✲❛♥❡❧✳ ❏á s❛❜❡♠♦s q✉❡ B é ✉♠❛ k✲á❧❣❡❜r❛✱ r❡st❛ ✈❡r✐✜❝❛r♠♦s q✉❡ s ❡ t sã♦ ♠♦r✲ ✜s♠♦s ❞❡ k✲á❧❣❡❜r❛✳ ▼❛s t❡♠♦s q✉❡ s ❡ t sã♦ k✲❧✐♥❡❛r❡s ❡ s ♠♦r✜s♠♦ ❞❡ k

  ✲á❧❣❡❜r❛✱ ♦✉ s❡❥❛✱ ❜❛st❛ ✈❡r✐✜❝❛r♠♦s q✉❡ t é ❛♥t✐♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ❆♥t❡s ✈❛♠♦s ♠♦str❛r ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ♥❡❝❡ssár✐❛s✳ ◆♦t❡ q✉❡ ♣❡❧♦ ❢❛t♦ ❞❡ ∆ s❡r ♠✉❧t✐♣❧✐❝❛t✐✈♦ t❡♠♦s✱ ♣❛r❛ t♦❞♦ b ∈ B✱ b ⊗ b = ∆(b) = ∆(b1 B ) = ∆(b)∆(1 B ) = b

  1 ⊗ b 1 .

  

(1) (2) (1) (1) (2) (2)

  ✭✷✳✶✶✮ ❆❣♦r❛✱ ✉s❛♥❞♦ ❡ss❡ ❢❛t♦✱ t❡♠♦s ♣❛r❛ t♦❞♦ b ∈ B✱ R

  ⊓ (b ) ⊗ b = 1 ε(b 1 ) ⊗ b

  (1) (2) (1) (1) (2) (2) ′ ′

  = 1 ε(b

  1 1 ) ⊗ b

  1

  (1) (1) (1 ) (2) (2) (2 ) ✭♣♦r ✷✳✶✶✮

  = 1 ε(b 1 ) ⊗ b

  1

  (1) (1) (2) (2) (3) ✭♣♦r ✷✳✾✮

  = 1 ε(b 1 ) ⊗ b

  1

  (1) (1) (2)(1) (2) (2)(2)

  = 1 ε((b1 ) ) ⊗ (b1 )

  (1) (2) (1) (2) (2)

  = 1 ⊗ ε((b1 ) )(b1 )

  (1) (2) (1) (2) (2) = 1 ⊗ b1 . (1) (2) ✭✷✳✶✷✮

  ❉❡ss❛ ❢♦r♠❛ t❡♠♦s R R 1 ⊗ ⊓ (1 ) ⊗ 1 = 1 ⊗ ⊓ (1 ) ⊗ 1

  

(1) (2) (3) (1) (2)(1) (2)(2)

′ ′

  = 1 ⊗ 1 ⊗ 1 1 .

  (1) (1 ) (2) (2 ) ✭✷✳✶✸✮

  P♦rt❛♥t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ R✱ t❡♠♦s ′ ′ t(b)t(a) = ε(b1 )1 ε(a1 )1

  (1) (2) (1 ) (2 ) ′ ′

  = ε(b1 )ε(a1 )1

  1

  (1) (1 ) (2) (2 ) R

  = ε(b1 )ε(a ⊓ (1 ))1

  (1) (2) (3) ′ ′

  = ε(b1 )ε(a1 ε(1 1 ))1

  (1) (1 ) (2) (2 ) (3) ✭♣♦r ✷✳✶✸✮ ′ ′

  = ε(b1 )ε(1 1 )ε(a1 )1

  

(1) (2) (2 ) (1 ) (3)

′ ′

  = ε(b1 1 )ε(a1 )1

  (1) (2 ) (1 ) (2) ✭♣♦r ✷✳✶✵✮

  = ε(b1 )ε(a1 )1

  

(2) (1) (3) ✭♣♦r ✷✳✾✮

  = ε(bε(a1 )1 )1

  (1) (2) (3) ′ ′

  = ε(bε(a1 )1 1 )1

  (1) (2) (1 ) (2 ) ✭♣♦r ✷✳✾✮ ′ ′

  = ε(bt(a)1 )1

  (1 ) (2 )

  = t(bt(a)), ✭✷✳✶✹✮ t❛♠❜é♠ t❡♠♦s t(t(a)b) = t(ε(a1 )1 )

  (1) (2)

  = ε(a1 )t(1

  b)

  

(1) (2)

′ ′

  = ε(a1 )ε(1 b1 )1

  (1) (2) (1 ) (2 ) ′ ′

  = ε(ab1 )1

  (1 ) (2 ) ✭♣♦r ✷✳✶✵✮ = t(ab).

  ❆❣♦r❛ ♥♦t❡ q✉❡ R ′ ′ ⊓ (b)t(a) = 1 ε(b1 )ε(a1 )1

  (1) (2) (1 ) (2 ) ′ ′

  = ε(a1 )1 1 ε(b1 )

  (1 ) (1) (2 ) (2) ′ ′

  = ε(a1 )1 1 ε(b1 )

  (1 ) (2 ) (1) (2)

R

= t(a) ⊓ (b).

P♦rt❛♥t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ R✱ t❡♠♦s t(ab) = t(t(a)b)

  R = t(t(a) ⊓ (b)) R

  = t(⊓ (b)t(a)) = t(bt(a)) = t(b)t(a). e ✭♣♦r ✷✳✶✹✮

  ❙❡❣✉❡ ❞❛í q✉❡ B é ✉♠ R ✲❛♥❡❧✳ ❱❛♠♦s ✈❡r✐✜❝❛r ❛❣♦r❛ q✉❡ (B, ∆, ε) é ✉♠ R✲❝♦❛♥❡❧✳ ❉❡ ❢❛t♦✱ ❥á t❡♠♦s q✉❡ ❛ ❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡ é s❛t✐s❢❡✐t❛✳ ❱❛♠♦s ✈❡r✐✜❝❛r ♦ ❛①✐♦♠❛ ❞❛ ❝♦✉♥✐❞❛❞❡✳ ❉❡ ❢❛t♦✱

  (ε ⊗ B)∆(b) = (ε ⊗ R B)(b ⊗ R b )

  (1) (2)

  = ε(b ) · b

  

(1) (2)

  = 1 ε(b 1 ) · b

  (1) (1) (2) (2)

  = b t(1 ε(b 1 ))

  (2) (1) (1) (2)

  = b ε(b 1 )t(1 )

  (2) (1) (2) (1) ′ ′

  = b ε(b 1 )ε(1 1 )1

  (2) (1) (2) (1) (1 ) (2 ) ′ ′

  = b ε(b 1 )1

  (2) (1) (1 ) (2 ) ✭♣♦r ✷✳✶✵✮ ′ ′

  = ε(b 1 )b

  1

  (1) (1 ) (2) (2 )

  = ε(b )b = b (1) (2) ✭♣♦r ✷✳✶✶✮. ❚❛♠❜é♠✱

  (B ⊗ ε)∆(b) = (B ⊗ ε)(b ⊗ R b )

  (1) (2)

  = b · ε(b )

  

(1) (2)

  = b s(ε(b ))

  

(1) (2)

′ ′

  = b s(1 ε(b 1 ))

  (1) (1 ) (2) (2 ) ′ ′

  = b 1 ε(b 1 )

  (1) (1 ) (2) (2 )

  = b ε(b ) = b (1) (2) ✭♣♦r ✷✳✶✶✮. ❱❛♠♦s ✈❡r ❛❣♦r❛ q✉❡ ∆ ❡ ε sã♦ ♠♦r✜s♠♦s ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R ❡ b ∈ B✱ t❡♠♦s

  ∆(r · b) = ∆(bt(r)) = ∆(bε(r1 )1 )

  (1) (2)

  = ε(r1 )∆(b1 )

  

(1) (2)

  = ε(r1 )(b 1 ⊗ R b 1 ),

  (1) (1) (2) (2) (3)

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ r · ∆(b) = r · (b ⊗ R b )

  (1) (2)

  = r · b ⊗ R b

  (1) (2)

  = b

  (2)

  (2 )

  = 1

  (1)

  b

  (1)

  ⊗ ε(1

  (1 )

  1

  b

  (2)

  (2)

  )1

  (2 ) ✭♣♦r ✷✳✶✶✮

  = 1

  (1)

  b

  (1)

  ⊗ ε(1

  )1

  b

  b

  (1)

  (2)

  ⊓ R (r)

  2.11 = ∆(b) · r,

  ❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ é ❞❡✈✐❞♦ ❛♦ ❢❛t♦ q✉❡ ⊓ R é ✐❞❡♠♣♦t❡♥t❡✱ ♣♦✐s ❝♦♠♦ r ∈ R ✱ t❡♠♦s q✉❡ ❡①✐st❡ a ∈ B✱ t❛❧ q✉❡ ⊓ R

  (a) = r ✱ ❞❡ss❛ ❢♦r♠❛

  ⊓ R (r) = ⊓ R (⊓ R (a)) = ⊓ R (a) = r.

  ✭✷✳✶✺✮ P♦rt❛♥t♦✱ ∆ é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❈♦♥s✐❞❡r❡ ❛❣♦r❛✱ ♣❛r❛ t♦❞♦ b ∈ B

  ✱ ❛ s❡❣✉✐♥t❡ ❛♣❧✐❝❛çã♦ ⊓ L : B −→ B, b 7−→ ε(1

  b)1

  (1

)

  (2) .

  ❙❡❣✉❡ q✉❡ b

  (1)

  ⊗ ⊓ L (b

  (2)

  ) = b

  (1)

  ⊗ ε(1

  (2)

  (2)

  (1)

  (2)

  (1)

  b)

  (1)

  ε((1

  (1)

  b)

  (2)

  ) ⊗ 1

  = 1

  (2)

  (1)

  b ⊗ 1

  (2) .

  ✭✷✳✶✻✮ ❚❡♠♦s ❡♥tã♦✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ B✱

  ⊓ R (a)b = 1

  (1)

  ε(a1

  

(2)

  = (1

  )1

  )1

  (2)

  (3) ✭♣♦r ✷✳✾✮

  = 1

  (1)(1)

  b

  (1)

  ⊗ ε(1

  ((1)(2)

  b

  )1

  (2)

  (2)

  = (1

  (1)

  b)

  (1)

  ⊗ ε((1

  (1)

  b)

  ⊗ R b

  ⊓ R (r) = b

  (1)

  1

  = b

  (1)

  1

  (2)

  ε(r1

  (1)

  ) ⊗ R b

  (2)

  (3) ✭♣♦r ✷✳✾✮

  1

  = ε(r1

  (1)

  )(b

  (1)

  1

  (2)

  ⊗ R b

  (2)

  (2 ) ✭♣♦r ✷✳✶✶✮

  (2)

  (3) ).

  ⊗ R b

  t(r) ⊗ R b

  (2)

  = b

  (1)

  ε(r1

  (1)

  )1

  (2)

  (2)

  ) ⊗ R b

  = b

  (1)

  1

  (1 )

  1

  (2)

  ε(r1

  

(1)

  1

  ❚❛♠❜é♠ t❡♠♦s✱ ∆(b · r) = ∆(b ⊓ R (r))

  

(2)

  ε(r1

  1

  (1)

  ⊗ R b

  (2)

  1

  

(2)

  1

  (1 )

  (2 )

  ) = b

  ) ✭♣♦r ✷✳✾✮

  = b

  (1)

  1

  (1)

  ⊗ R b

  (2)

  1

  (1)

  (3)

  = ∆(b)∆(⊓ R (r)) = (b

  ε(r1

  (1)

  ⊗ R b

  (2)

  )(1

  

(1)

  ⊗ R

  1

  (2)

  (3)

  ε(r1

  )) = b

  (1)

  1

  (1)

  ⊗ R b

  (2)

  1

  

(2)

  )b

  = 1

  ⊗ ⊓ R (b)1

  (R)

  ε(a(⊓

  (1)

  ) = (⊓ R (b))

  (2)

  ε(a ⊓ R (b)1

  (1)

  ⊓ R (a ⊓ R (b)) = 1

  ✭✷✳✶✽✮ ❙❡❣✉❡ q✉❡✱

  (2) .

  (1)

  (2)

  = 1

  (2)

  )1

  (2 )

  ε(b1

  (1

)

  ⊗ 1

  (1)

  = 1

  ) ✭♣♦r ✷✳✾✮

  (2 )

  (b))

  ) ✭♣♦r ✷✳✶✽✮

  (2)

  (2)

  )

  (2 )

  1

  (2)

  ε(b1

  (1 )

  )1

  (1)

  ) = ε(r1

  (2 )

  1

  )1

  = ⊓ R (a) ⊓ R (b).

  (1)

  ε(bε(r1

  (1 )

  ) = 1

  (2

)

  ε(bt(r)1

  (1 )

  = ⊓ R (bt(r)) = 1

  P♦r ♦✉tr♦ ❧❛❞♦ ε(r · b) = ⊓ R (r · b)

  ε(b · r) = ⊓ R (b · r) = ⊓ R (br) = ⊓ R (b ⊓ R (r)) = ⊓ R (b) ⊓ R (r) = ⊓ R (b)r = ε(b)r.

  ✭♣♦r ✷✳✶✼✮ ✭✷✳✶✾✮

  ε(b1

  1

  (1)

  (2)

  (2)

  b

  (1)

  )ε(1

  

(2)

  ε(a1

  (1)

  ) = b

  (2)

  )1

  b

  (1)

  (1)

  ε(aε(1

  (1)

  = b

  )) ✭♣♦r ✷✳✶✻✮

  (2)

  ε(a ⊓ L (b

  (1)

  ) = b

  

(2)

  bε(a1

  ) = b

  ε(ab

  (1

)

  (1)(2)

  ⊗ 1

  (1)

  ) = 1

  (3)

  ε(b1

  (2)

  ⊗ 1

  (1)

  ) = 1

  (2)

  ε(b1

  ⊗ 1

  

(2)

  (1)(1)

  ) = 1

  (2)

  )ε(b1

  (1)

  ) = ∆(1

  

(2)

  ε(b1

  (1)

  ❆❣♦r❛ ♥♦t❡ q✉❡ ∆(⊓ R (b)) = ∆(1

  ) ✭♣♦r ✷✳✶✵✮. ✭✷✳✶✼✮

P♦rt❛♥t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R ❡ b ∈ B✱ t❡♠♦s

  = ε(r1

  = ε(a1

  (2)

  )ε(b

  (1)

  )1

  (3)

  b

  (3)

  (1)

  (2)

  )ε(1

  (2)

  b

  (1)

  )1

  (3)

  b

  (2)

  b

  )ε(1

  (1)

  (1)

  (1)

  )1

  (3)

  b

  (2) ✭♣♦r ✷✳✶✸✮

  = ε(a1

  (1)

  )ε(b

  ε(1

  (1)

  

(2)

  b

  (2)

  ))1

  (3)

  b

  (3) ✭♣♦r ✷✳✶✼✮

  = ε(a1

  = ε(a1

  b

  (2)

  ⊗ R t(r)b (2) ✭♣♦r ✷✳✷✵✮. ❙❡❣✉❡ q✉❡ ∆ ❡stá ♥♦ ❚❛❦❡✉❝❤✐ à ❞✐r❡✐t❛✳ ❱❛♠♦s ✈❡r ❛❣♦r❛ q✉❡ ε é ✉♠ ❝❛r❛❝t❡r à ❞✐r❡✐t❛ ❡♠ (B, s)✳ ❉❡ ❢❛t♦✱ ♥♦t❡ q✉❡

  = b

  (1)

  ⊗ R ε(rb

  (2)

  )b

  (3)

  = b

  (1)

  ε(1 B ) = ⊓ R (1 B ) = 1

  ) ⊗ R b

  (1)

  ε(1 B

  1

  (2)

  ) = 1

  (1)

  ε(1

  (2)

  (3) ✭♣♦r ✷✳✶✼✮

  (2)

  (1)

  P♦rt❛♥t♦✱ s(r)b

  )1

  (2)

  b

  

(2)

  = ε(ab

  (1)

  )b

  (2) ✭♣♦r ✷✳✶✶✮. ✭✷✳✷✵✮

  (1)

  ε(rb

  ⊗ R b

  (2)

  = ⊓ R (r)b

  (1)

  ⊗ R b

  (2)

  = b

  (1)

  )b

  )ε(⊓ R (1

  (1)

  (1)

  ) ⊓ R (1

  

(2)

  ⊓ R (b)) = ε(r1

  (1)

  ) ⊓ R (1

  

(2)

  ) ⊓ R (b) ✭♣♦r ✷✳✶✾✮

  = ε(r1

  )1

  = ε(r1

  (1 )

  ε(1

  

(2)

  1

  (2 )

  ) ⊓ R (b) = 1

  (1 )

  ε(r1

  (1)

  )) ✭♣♦r ✷✳✾✮

  )ε(1

  

(2)

  ) ⊓ R (1

  

(2)

  )ε(b1

  (3)

  ) ✭♣♦r ✷✳✶✸✮

  = ε(r1

  (1)

  ) ⊓ R (1

  ε(b1

  (2 )

  (3)

  )) = ε(r1

  (1)

  ) ⊓ R (1

  

(2)

  1

  (1 )

  ε(b1

  (1)

  

(2)

  (1)

  (1 )

  (1)

  )1

  (2)

  b

  (2)

  = ε(a1

  (1)

  )ε(1

  b

  (1)

  (1)

  )1

  (2)

  1

  (2 )

  b

  (2) ✭♣♦r ✷✳✶✶✮

  = ε(a1

  )ε(b

  = ε(a1

  1

  (1)

  (2 )

  ) ⊓ R (b) = 1

  (1 )

  ε(r1

  (2 )

  ) ⊓ R (b) ✭♣♦r ✷✳✶✵✮

  = ⊓ R (r) ⊓ R (b) = r ⊓ R (b) = rε(b). ❙❡❣✉❡ q✉❡ ε é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✳ P♦rt❛♥t♦✱ ❝♦♥❝❧✉í♠♦s ❛ss✐♠ q✉❡ (B, ∆, ε) é ✉♠ R✲❝♦❛♥❡❧✳ ❱❛♠♦s ✈❡r✐✜❝❛r q✉❡ ❛ ✐♠❛❣❡♠ ❞❡ ∆ ❡stá ❝♦♥t✐❞❛ ❡♠ B × r R

  B ✳ ❉❡ ❢❛t♦✱ ❛♥t❡s t❡♠♦s✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ B✱ t(a)b = ε(a1

  )1

  (2)

  (2)

  b = ε(a1

  (1)

  )1

  (2)

  ε(b

  (1)

  )b

  ) = 1 B , t❡♠♦s t❛♠❜é♠✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ B✱ ε(s(ε(a))b) = ε(ε(a)b) R R

  = ⊓ (⊓ (a)b) R = ⊓ (1 ε(a1 )b)

  (1) (2) R

  = ε(a1 ) ⊓ (1

  b) ′ ′ (2) (1) = 1 ε(a1 )ε(1 b1 )

  (1 ) (2) (1) (2 )

′ ′

  = 1 ε(ab1 )

  (1 ) (2 ) ✭♣♦r ✷✳✶✵✮ R = ⊓ (ab) = ε(ab).

  P♦rt❛♥t♦✱ ❝♦♥❝❧✉í♠♦s ❛ss✐♠✱ q✉❡ B é ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❊①❡♠♣❧♦ ✷✳✶✷ ❈♦♥s✐❞❡r❡ H ✉♠❛ k✲❜✐á❧❣❡❜r❛ ❝♦♠✉t❛t✐✈❛ ❡ R ✉♠ H✲ ❝♦♠ó❞✉❧♦ á❧❣❡❜r❛ à ❞✐r❡✐t❛ ❝♦♠✉t❛t✐✈♦✳ ❊♥tã♦✱ t❡♠♦s q✉❡ ♦ ♣r♦❞✉t♦ k H t❡♥s♦r✐❛❧ R⊗ ✱ ❝♦♠ ❡str✉t✉r❛ ❞❡ á❧❣❡❜r❛ ♣❛❞rã♦✱ é ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❖s ♠♦r✜s♠♦s s♦✉r❝❡ ❡ t❛r❣❡t sã♦ ❞❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱

  (0) (1)

  ⊗ a = ρ(a) ♣❛r❛ t♦❞♦ a ∈ R✱ ♣♦r s(a) = a ✱ ♦✉ s❡❥❛✱ ♦ s♦✉r❝❡ é H ❞❛❞♦ ♣❡❧❛ ❝♦❛çã♦ ❞❡ H ❡♠ R ❡ t(a) = a ⊗ 1 ✳ ❚❡♠♦s t❛♠❜é♠ q✉❡ k H −→ (R ⊗ k

  H) ⊗ R (R ⊗ k

  H) ♦ ❝♦♣r♦❞✉t♦ ∆ : R ⊗ é ❞❡✜♥✐❞♦ ♣♦r ∆(a ⊗ h) = (a ⊗ h ) ⊗ R (1 R ⊗ h ) k H −→ R

  (1) (2) ❡ ❛ ❝♦✉♥✐❞❛❞❡ ε : R ⊗ k H

  é ❞❛❞❛ ♣♦r ε(a ⊗ h) = aε(h)✱ ♣❛r❛ t♦❞♦ a ⊗ h ∈ R ⊗ ✳ ➱ ❝❧❛r♦ q✉❡ ❛s ✐♠❛❣❡♥s ❞❡ s ❡ t ❝♦♠✉t❛♠ ❡ q✉❡ sã♦ ♠♦r✜s♠♦s ❞❡ k✲ á❧❣❡❜r❛ ✭♣❡❧❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❡ R ❡ H✮✱ ♦✉ s❡❥❛✱ t é ❛♥t✐♠♦r✜s♠♦ ❞❡ op k k H

  ✲á❧❣❡❜r❛s✳ P♦rt❛♥t♦✱ R ⊗ é ✉♠ R ⊗ R ✲❛♥❡❧✳ ◆♦t❡ q✉❡ r (R ⊗ k

  H) ⊗ R (R ⊗ k

  H) = (R ⊗ k R (b ⊗ k) ∈ (R ⊗ k

  H) × (R ⊗ k R H).

  H) ⊗ R (R ⊗ k

  H) ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r (a ⊗ h) ⊗ ❡ r ∈ R t❡♠♦s

  (0) (1)

  s(r)(a ⊗ h) ⊗ R (b ⊗ k) = (r ⊗ r )(a ⊗ h) ⊗ R (b ⊗ k)

  (0) (1)

  = (a ⊗ h)(r ⊗ r ) ⊗ R (b ⊗ k) = (a ⊗ h)s(r) ⊗ R (b ⊗ k) R = (a ⊗ h) · r ⊗ (b ⊗ k) = (a ⊗ h) ⊗ R r · (b ⊗ k) = (a ⊗ h) ⊗ R (b ⊗ k)t(r) = (a ⊗ h) ⊗ R (b ⊗ k)(r ⊗ 1 H ) = (a ⊗ h) ⊗ R (r ⊗ 1 H )(b ⊗ k)

  k H = (a ⊗ h) ⊗ R t(r)(b ⊗ k). ❱❛♠♦s ✈❡r ❛❣♦r❛ q✉❡ R ⊗ é ✉♠ R✲❝♦❛♥❡❧✳ ❉❡ ❢❛t♦✱ ♣r✐♠❡✐r♦ ✈❛✲ ♠♦s ✈❡r✐✜❝❛r q✉❡ ∆ ❡ ε sã♦ ♠♦r✜s♠♦s ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ k

  H) ⊗ R (R ⊗ k

  H) q✉❛✐sq✉❡r a ⊗ h ∈ (R ⊗ ❡ b ∈ R✱ t❡♠♦s

  (0) (1)

  ∆((a ⊗ h) · b) = ∆(ab ⊗ hb )

  (0) (1) (1)

  = (ab ⊗ (hb ) ) ⊗ R (1 R ⊗ (hb ) )

  (1) (2) (1) (1) (0)

  = (ab ⊗ h b ) ⊗ R (1 R ⊗ h b )

  (1) (2) (1) (2) (0)(0) (0)(1) (1)

  = (ab ⊗ h b ) ⊗ R (1 R ⊗ h b )

  (1) (2) (0) (1)

  = (a ⊗ h )s(b ) ⊗ R (1 R ⊗ h b )

  (1) (2) (0) (1)

  = (a ⊗ h ) · b ⊗ R (1 R ⊗ h b )

  (1) (2) (0) (1)

  = (a ⊗ h ) ⊗ R b · (1 R ⊗ h b )

  (1) (2) (1) (0)

  = (a ⊗ h ) ⊗ R (1 R ⊗ h b )t(b )

  (1) (2) (1) (0)

  = (a ⊗ h ) ⊗ R (1 R ⊗ h b )(b ⊗ 1 H )

  (1) (2) (0) (1)

  = (a ⊗ h ) ⊗ R (b ⊗ h b )

  (1) (2) (0) (1)

  = (a ⊗ h ) ⊗ R (1 R ⊗ h )(b ⊗ b )

  (1) (2)

  = (a ⊗ h ) ⊗ R (1 R ⊗ h )s(b)

  (1) (2)

  = (a ⊗ h ) ⊗ R (1 R ⊗ h ) · b

  (1) (2) = ∆(a ⊗ h) · b.

  ✭✷✳✷✶✮ P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s

  ∆(b · (a ⊗ h)) = ∆((a ⊗ h)t(b)) = ∆((a ⊗ h)(b ⊗ 1 H )) = ∆(ab ⊗ h) = (ab ⊗ h ) ⊗ R (1 R ⊗ h )

  (1) (2)

  = (a ⊗ h )(b ⊗ 1 H ) ⊗ R (1 R ⊗ h )

  (1) (2)

  = (a ⊗ h )t(b) ⊗ R (1 R ⊗ h )

  (1) (2)

  = b · (a ⊗ h ) ⊗ R (1 R ⊗ h )

  (1) (2) = b · ∆(a ⊗ h).

  ❚❛♠❜é♠ t❡♠♦s

  (0) (1)

  ε((a ⊗ h) · b) = ε(ab ⊗ hb )

  = ab

  ) ⊗ R (1 R ⊗ h

  ) = (aε(h

  (2)

  ) · (1 R ⊗ h

  

(1)

  )) = ε(a ⊗ h

  (2)

  (1)

  )) · (1 R ⊗ h

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s (ε ⊗ B)∆(a ⊗ h) = (ε ⊗ B)((a ⊗ h

  ) = a ⊗ h,

  (2)

  ε(h

  (1)

  ) = a ⊗ h

  (1)

  (2)

  ) · 1 R ε(h

  (1)

  (2)

  ) ⊗ R (1 R ⊗ k

  (1)

  ))((b ⊗ k

  (2)

  ) ⊗ R (1 R ⊗ h

  ❱❛♠♦s ✈❡r q✉❡ ∆ é ♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s✳ P❛r❛ t❛♥t♦✱ ❜❛st❛ ♠♦str❛r q✉❡ ∆ é ♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a ⊗ h, b ⊗ k ∈ R ⊗ H✱ t❡♠♦s ∆(a ⊗ h)∆(b ⊗ k) = ((a ⊗ h

  ) = (1 R ⊗ h

  (2) = a ⊗ h.

  ) ⊗ h

  (1)

  ) ⊗ 1 H ) = aε(h

  (1)

  )(aε(h

  

(2)

  (2)

  (1)

  (0)

  (2)

  (2)(2)

  ) ⊗ R (1 R ⊗ h

  (2)(1)

  ) ⊗ R (1 R ⊗ h

  (1)

  )) = (a ⊗ h

  ) ⊗ R (1 R ⊗ h

  (1)(1)

  (1)

  (B ⊗ ∆)∆(a ⊗ h) = (B ⊗ ∆)((a ⊗ h

  = abε(h) = bε(a ⊗ h). ❆❣♦r❛ ✈❛♠♦s ✈❡r✐✜❝❛r ❛ ❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ ❛ ❝♦✉♥✐t❛❧✐❞❛❞❡✳ P❛r❛ ❢❛❝✐✲ ❧✐t❛r✱ ❞❡♥♦t❡ B = R ⊗ k H ✳ ❊♥tã♦✱ ♣❛r❛ q✉❛❧q✉❡r a ⊗ h ∈ B✱ t❡♠♦s

  ❡ ε(b · (a ⊗ h)) = ε(ab ⊗ h)

  ) = aε(h)b = ε(a ⊗ h)b

  (1)

  ε(hb

  ) = (a ⊗ h

  ) ⊗ R (1 R ⊗ h

  ) = (a ⊗ h

  (1)

  (2)

  ) · ε(1 R ⊗ h

  (1)

  )) = (a ⊗ h

  (2)

  ) ⊗ R (1 R ⊗ h

  (B ⊗ ε)∆(a ⊗ h) = (B ⊗ ε)((a ⊗ h

  (1)(2)

  ) = (∆ ⊗ B)∆(a ⊗ h). ❚❛♠❜é♠ t❡♠♦s

  (2)

  ) ⊗ R (1 R ⊗ h

  (1)

  ) = ∆(a ⊗ h

  (2)

  ) ⊗ R (1 R ⊗ h

  ))

  = (a ⊗ h )(b ⊗ h ) ⊗ R (1 R ⊗ h )(1 R ⊗ k )

  (1) (1) (2) (2)

  = (ab ⊗ h k ) ⊗ R (1 R ⊗ h k )

  (1) (1) (2) (2)

  = (ab ⊗ (hk) ) ⊗ R (1 R ⊗ (hk) )

  (1) (2) = ∆(ab ⊗ hk) = ∆((a ⊗ h)(b ⊗ k)).

  ❘❡st❛ ♠♦str❛r♠♦s q✉❡ ε é ✉♠ ❝❛r❛❝t❡r à ❞✐r❡✐t❛ ♥♦ R✲❛♥❡❧ (R ⊗ H, s)✳ R ⊗1 H ) = 1 R ❉❡ ❢❛t♦✱ ❝❧❛r♦ q✉❡ ε(1 ❡ ❝♦♠♦ ε é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s t❡♠♦s ε((a ⊗ h)s(b)) = ε(a ⊗ h)b✳ ❆❣♦r❛✱ ♣❛r❛ q✉❛✐sq✉❡r a ⊗ h, b ⊗ k ∈ R ⊗ H t❡♠♦s

  ε(s(ε(a ⊗ h))(b ⊗ k)) = ε(s(aε(h))(b ⊗ k))

  (0) (1)

  = ε(ε(h)(a ⊗ a )(b ⊗ k))

  (0) (1)

  = ε(ε(h)(a b ⊗ a k))

  (0) (1)

  = ε(a bε(h) ⊗ a k)

  (0) (1)

  = a bε(h)ε(a k)

  (0) (1)

  = a ε(a )bε(hk) = abε(hk) = ε(ab ⊗ hk) k H = ε((a ⊗ h)(b ⊗ k)).

  ❈♦♥❝❧✉í♠♦s ❛ss✐♠ q✉❡ R ⊗ é ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❊①❡♠♣❧♦ ✷✳✶✸ ❙❡❥❛ H ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢ ❝♦❝♦♠✉t❛t✐✈❛ ❡ R ✉♠ H✲ ♠ó❞✉❧♦ á❧❣❡❜r❛ à ❡sq✉❡r❞❛ ❝♦♠✉t❛t✐✈♦✳ ❊♥tã♦ ♦ ♣r♦❞✉t♦ s♠❛s❤ R#H✱ k H ♦ q✉❛❧ é ✉♠❛ k✲á❧❣❡❜r❛✱ ✐s♦♠♦r❢❛ ❛ R ⊗ ❝♦♠♦ k✲♠ó❞✉❧♦✱ ❝♦♠ ♠✉❧✲ t✐♣❧✐❝❛çã♦✱ ❞❛❞❛ ♣❛r❛ t♦❞♦ (r#h), (s#k) ∈ R#H✱ ♣♦r (r#h)(s#k) = r(h ⊲ s)#h k R #1 H

  (1) (2) ❡ ✉♥✐❞❛❞❡ 1 ✱ é ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❖s H

  ♠♦r✜s♠♦s s♦✉r❝❡ ❡ t❛r❣❡t sã♦ ✐❣✉❛✐s ❛ s(r) = t(r) = r#1 ✳ ➱ ❝❧❛r♦ q✉❡ s ❡ t sã♦ ♠♦r✜s♠♦s ❞❡ k✲á❧❣❡❜r❛ ❡ q✉❡ ❝♦♠✉t❛♠ ♥❛ ✐♠❛✲ ❣❡♠✱ ♣♦✐s R é ❝♦♠✉t❛t✐✈♦✳ ❙❡❣✉❡ q✉❡ t á ❛♥t✐♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s✳ op P♦rt❛♥t♦✱ t❡♠♦s ✉♠❛ ❡str✉t✉r❛ ❞❡ R ⊗ R ✲❛♥❡❧ ❡♠ R#H✳ ❉❡♥♦t❛✲ ♠♦s ❛ ❛çã♦ ❞❡ H ❡♠ R✱ ♣♦r h ⊲ r ❡ ❛s ❛çõ❡s ❞❡ R ❡♠ R#H✱ ♣♦r

  ′ ′

  r · (a#k) · r ∈ R ✱ ♣❛r❛ q✉❛✐sq✉❡r r, r ✱ h ∈ H ❡ a#k ∈ R#H✳ ❱❛✲

  ♠♦s ❞❡✜♥✐r ✉♠❛ ❡str✉t✉r❛ ❞❡ R✲❝♦❛♥❡❧ ❡♠ R#H ❞❡✜♥✐♥❞♦ ♦ ❝♦♣r♦❞✉t♦ ∆ : R#H −→ (R#H)⊗ R (R#H) )⊗ R (1 R #h )

  ♣♦r ∆(r#h) = (r#h (1) (2) ❡ ❝♦✉♥✐❞❛❞❡ ε : R#H −→ R ❞❡✜♥✐❞❛ ♣♦r ε(r#h) = S(h) ⊲ r✱ ♣❛r❛ t♦❞♦ r#h ∈ R#H

  ✳

  ❱❛♠♦s ✈❡r q✉❡ ∆ ❡ ε sã♦ ♠♦r✜s♠♦s ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ❛♥t❡s ♥♦t❡ q✉❡ só ♣r❡❝✐s❛♠♦s ♠♦str❛r ♣♦r ✉♠ ❧❛❞♦✱ ♣♦✐s ❛s ❛çõ❡s✱ à ❞✐r❡✐t❛ ❡ à ❡sq✉❡r❞❛✱ ❞❡ R ❡♠ R#H sã♦ ✐❣✉❛✐s✱ ❞❡✈✐❞♦ ❛♦ ❢❛t♦ q✉❡ ♦s ♠♦r✜s♠♦s s♦r❝❡ ❡ t❛r❣❡t sã♦ ✐❣✉❛✐s✱ ❡ t❛♠❜é♠ R é ❝♦♠✉❛t✐✈♦✳ P♦rt❛♥t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a#h ∈ R#H ❡ r ∈ R t❡♠♦s

  ∆((a#h) · r) = ∆((a#h)(r#1 H )) = ∆(a(h ⊲ r)#h )

  (1) (2)

  = (a(h ⊲ r)#h ) ⊗ R (1 R #h )

  (1) (2)(1) (2)(2)

  = (a(h ⊲ r)#h ) ⊗ R (1 R #h )

  (1)(1) (1)(2) (2)

  = (a#h )(r#1 H ) ⊗ R (1 R #h )

  (1) (2)

  = (a#h ) · r ⊗ R (1 R #h )

  (1) (2)

  = (a#h ) ⊗ R r · (1 R #h )

  (1) (2)

  = (a#h ) ⊗ R (1 R #h ) · r

  (1) (2) = ∆(a#h) · r.

  ❚❛♠❜é♠ t❡♠♦s ε((a#h) · r) = ε(a(h ⊲ r)#h )

  (1) (2)

  = S(h ) ⊲ (a(h ⊲ r))

  (2) (1)

  = (S(h ) ⊲ a)(S(h )h ⊲ r)

  (3) (2) (1)

  = (S(h ) ⊲ a)(S(h )h ⊲ r)

  (3) (1) (2)

  = (S(h ) ⊲ a)(ε(h )1 R r)

  (2) (1)

  = (S(h) ⊲ a)r = ε(a#h)r. ❆❣♦r❛ ♥♦t❡ q✉❡ ♦ ❝♦♣r♦❞✉t♦✱ ❞❡✜♥✐❞♦ ♥❡ss❡ ❡①❡♠♣❧♦✱ é ♦ ♠❡s♠♦ ❞♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✳ P♦rt❛♥t♦✱ ❜❛st❛ ♠♦str❛r♠♦s ❛ ❝♦✉♥✐t❛❧✐❞❛❞❡ ♣❛r❛ ♦❜t❡r♠♦s ✉♠❛ ❡str✉t✉r❛ ❞❡ R✲❝♦❛♥❡❧ ❡♠ R#H✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ a#h ∈ R#H

  ✱ ❞❡♥♦t❡ B = R#H✱ ❛ss✐♠ t❡♠♦s (B ⊗ ε)∆(a#h) = (B ⊗ ε)((a#h ) ⊗ R (1 R #h ))

  (1) (2)

  = (a#h ) · (S(h ) ⊲ 1 R )

  (1) (2)

  = (a#h ) · (ε(S(h ))1 R )

  (1) (2)

  = (a#h )ε(h )

  (1) (2) = a#h.

  ❚❛♠❜é♠ t❡♠♦s (ε ⊗ B)∆(a#h) = (ε ⊗ B)((a#h ) ⊗ R (1 R #h ))

  (1) (2)

  = (S(h

  (1)

  (1)

  ) ⊗ R (r#h

  (2)

  ) = (a#h

  (1)

  ) ⊗ R (r#1 H )(1 R #h

  (2)

  ) = (a#h

  ) ⊗ R t(r)(1 R #h

  (3)

  (2) ).

  ❱❛♠♦s ✈❡r q✉❡ ∆ é ♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s✳ P❛r❛ t❛♥t♦✱ ❜❛st❛ ✈❡r✐✜✲ ❝❛r♠♦s q✉❡ ∆ é ♠✉❧t✐♣❧✐❝❛t✐✈♦✱ ♣♦✐s ❥á ♠♦str❛♠♦s q✉❡ ∆ é ♠♦r✜s♠♦ ❞❡ R

  ✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r (a#h), (r#k) ∈ R#H✱ t❡♠♦s ∆(a#h)∆(r#k) = ((a#h

  (1)

  ) ⊗ R (1 R #h

  (2)

  ))((r#k

  (1)

  ) = (a#h

  )r#h

  (2)

  (1)

  (1)

  ) ⊗ R (h

  (3)

  S(h

  (2)

  ) ⊲ r)#h

  (4)

  = (a#h

  ) ⊗ R (h

  (2)

  (2)

  S(h

  (3)

  ) ⊲ r#h

  (4)

  ) = (a#h

  (1)

  ) ⊗ R (ε(h

  ) ⊗ R (1 R #k

  )) = (a#h

  (2)

  ) = (a(h

  k

  (1)

  ) ⊗ R (ε(h

  (3)

  )1 R #h

  (4)

  k

  (2)

  (1)

  ⊲ r)#h

  ⊲ r)#h

  

(2)

  k

  (1)

  ) ⊗ R (1 R #h

  (3)

  k

  (2)

  

(2)

  (1)

  (1)

  ⊲ r)#h

  )(r#k

  

(1)

  ) ⊗ R (1 R #h

  (2)

  )(1 R #k

  (2)

  ) = (a(h

  (1)

  

(2)

  ) = (a(h

  k

  (1)

  ) ⊗ R ((h

  (3)

  ⊲ 1 R )#h

  (4)

  k

  (2)

  )⊲r#1 H ) = (a#h

  )(S(h

  

(1)

  ) ⊗ R (1 R #h

  ❱❛♠♦s ✈❡r ❛❣♦r❛ q✉❡ ∆(B) ⊆ B × r R B

  ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a#h ∈ R#H

  ✱ r ∈ R✱ t❡♠♦s s(r)(a#h

  (1)

  ) ⊗ R (1 R #h

  (2)

  ) = (r#1 H )(a#h

  (1)

  (2)

  )1 R a)#h

  ) = (ra#h

  (1)

  ) ⊗ R (1 R #h

  (2)

  ) = (ra#h

  (1)

  ε(h

  (2)

  (2) = a#h.

  (1)

  (3)

  S(h

  ) ⊲ a) · (1 R #h

  (2)

  ) = (1 R #h

  

(2)

  )((S(h

  (1)

  ) ⊲ a)#1 H ) = (h

  (2)

  (1)

  = (ε(h

  ) ⊲ a)#h

  (3)

  = (h

  (1)

  S(h

  (2)

  ) ⊲ a)#h

  (3)

  )) ⊗ R (1 R #h

  ) = (a(ε(h

  (3)

  (3)

  (2)

  ) ⊲ r#1 H ) ⊗ R (1 R #h

  (3)

  ) = (a#h

  (1)

  ) · (S(h

  (2)

  ) ⊲ r) ⊗ R (1 R #h

  ) = (a#h

  (1)

  (1)

  ) ⊗ R (S(h

  (2)

  ) ⊲ r) · (1 R #h

  (3)

  ) = (a#h

  (1)

  ) ⊗ R (1 R #h

  )(S(h

  ) = (a#h

  (1)

  ) ⊲ r)#h

  )1 H ⊲ r)#h

  (2)

  ) ⊗ R (1 R #h

  (3)

  ) = (a(h

  

(1)

  S(h

  (2)

  (3)

  (4)

  ) ⊗ R (1 R #h

  (4)

  ) = (a(h

  

(1)

  ⊲(S(h

  (3)

  )⊲r))#h

  (2)

  )⊗ R (1 R #h

  ),

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ ∆((a#h)(r#k)) = ∆(a(h ⊲ r)#h k)

  (1) (2) = (a(h ⊲ r)#h k )) ⊗ R (1 R #h k ). (1) (2) (1) (3) (2)

  ❱❛♠♦s ✈❡r ❛❣♦r❛ q✉❡ ε é ❝❛r❛❝t❡r à ❞✐r❡✐t❛ ❡♠ (R#H, s)✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a#h, r#k ∈ R#H✱ t❡♠♦s ε(s(ε(a#h))(r#k)) = ε(s(S(h) ⊲ a)(r#k))

  = ε(((S(h) ⊲ a)#1 H )(r#k)) = ε((S(h) ⊲ a)r#k) = S(k) ⊲ ((S(h) ⊲ a)r) = (S(k )S(h) ⊲ a)(S(k ) ⊲ r)

  (2) (1)

  = (S(hk ) ⊲ a)(S(k ) ⊲ r),

  (2) (1)

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ ε((a#h)(r#k)) = ε(a(h ⊲ r)#h k)

  

(1) (2)

  = S(h k) ⊲ (a(h ⊲ r))

  

(2) (1)

  = (S(h k ) ⊲ a)(S(h k )h ⊲ r))

  (3) (2) (2) (1) (1)

  = (S(h k ) ⊲ a)(S(h k )h ⊲ r))

  (3) (2) (1) (1) (2)

  = (S(h k ) ⊲ a)(S(k )cS(h )h ⊲ r))

  (3) (2) (1) (1) (2)

  = (S(h k ) ⊲ a)(S(k )ε(h )1 R ⊲ r)

  (2) (2) (1) (1) = (S(hk ) ⊲ a)(S(k ) ⊲ r).

  (2) (1)

  ❙❡❣✉❡ ♣♦rt❛♥t♦✱ q✉❡ ε é ❝❛r❛❝t❡r à ❞✐r❡✐t❛✳ ❉❡ss❛ ❢♦r♠❛✱ ❝♦♥❝❧✉í♠♦s q✉❡ R#H é ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳

  ✷✳✷ ❉✉❛❧✐❞❛❞❡

  ❆♣r❡s❡♥t❛♠♦s ♥❛ ❙❡çã♦ ✶✳✺ ✉♠ ❡st✉❞♦ s♦❜r❡ ❛ ❞✉❛❧✐❞❛❞❡ ❡♥tr❡ R✲ ❛♥é✐s ❡ R✲❝♦❛♥é✐s✳ ◆❡st❛ s❡çã♦ ✈❛♠♦s ❡st✉❞❛r ✉♠ ♣♦✉❝♦ ❞❛ ❞✉❛❧✐❞❛❞❡ ❞❡ R✲❜✐❛❧❣❡❜ró✐❞❡s✳

  ❖s ❛①✐♦♠❛s ❞❡ ❜✐á❧❣❡❜r❛s sã♦ ❛✉t♦✲❞✉❛✐s✱ ♦✉ s❡❥❛✱ ♦s ❞✐❛❣r❛♠❛s q✉❡ ❡①♣r❡ss❛♠ ♦s ❛①✐♦♠❛s ❞❡ ✉♠❛ ❜✐á❧❣❡❜r❛✱ ♥ã♦ s❡ ❛❧t❡r❛♠ q✉❛♥❞♦ ❛s ✢❡✲ ❝❤❛s sã♦ ✐♥✈❡rt✐❞❛s✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞✐ss♦✱ s❡ ✉♠❛ ❜✐á❧❣❡❜r❛ B é ♣r♦❥❡t✐✈❛ ❡ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞❛ ❝♦♠♦ k✲♠ó❞✉❧♦✱ ❡♥tã♦ ♦ ❞✉❛❧ t❡♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❜✐á❧❣❡❜r❛ t❛♠❜é♠✱ ❛ q✉❛❧ é ❛ ❡str✉t✉r❛ tr❛♥s♣♦st❛ ❞❛ ❡s✲ tr✉t✉r❛ ❞❡ ❜✐á❧❣❡❜r❛ ❡♠ B✳ ❊♠ ❝♦♥tr❛st❡ ❝♦♠ ❡st❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡

  ❜✐á❧❣❡❜r❛s✱ ♦s ❛①✐♦♠❛s ❞❡ ❜✐á❧❣❡❜ró✐❞❡s ♥ã♦ sã♦ ❛✉t♦✲❞✉❛✐s ♥❡ss❡ s❡♥✲ t✐❞♦✳ P❡❧♦ q✉❡ ❡st✉❞❛♠♦s ♥❛ ❙❡çã♦ ✶✳✺✱ s❡ B é ✉♠ R✲❜✐á❧❣❡❜ró✐❞❡ ✭à ❞✐r❡✐t❛ ♦✉ à ❡sq✉❡r❞❛✮ ♣r♦❥❡t✐✈♦ ❡ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ❝♦♠♦ R✲♠ó❞✉❧♦ ✭à ❞✐r❡✐t❛ ♦✉ à ❡sq✉❡r❞❛✮✱ ♦ s❡✉ ❞✉❛❧ ✭à ❞✐r❡✐t❛ ♦✉ à ❡sq✉❡r❞❛✮ t❡♠ ❡str✉t✉r❛s ❞❡ R✲❛♥❡❧ ❡ R✲❝♦❛♥❡❧✳ ▼❛s ♥ã♦ é ❝❧❛r♦ q✉❡ ❡st❛s ❡str✉t✉✲ r❛s ❝♦♥st✐t✉❡♠ ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡✳ ❖ ♣r✐♠❡✐r♦ ❛ ♠♦str❛r ❡st❡ ❢❛t♦ ❢♦✐ ❙❝❤❛✉❡♥❜✉r❣ ❡♠ ❬✷✻❪✳ ❯♠ ❡st✉❞♦ ♠❛✐s ❞❡t❛❧❤❛❞♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ♥♦ ❛rt✐❣♦ ❬✶✺❪✳ ❙❡❣✉✐♠♦s ❛q✉✐ ♥♦ss❛ r❡❢❡rê♥❝✐❛ ♣r✐♥❝✐♣❛❧ ❬✼❪✱ q✉❡ ❡stá ♠❛✐s ♣ró①✐♠♦ ❞❡ ✭❬✶✺❪✱ Pr♦♣♦s✐çã♦ ✷✳✺✮✳ Pr♦♣♦s✐çã♦ ✷✳✶✹ ❙❡❥❛ B ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛✱ ♦ q✉❛❧ é ♣r♦✲ ❥❡t✐✈♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ❝♦♠♦ R✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ ✭✈✐❛ ❛ ♠✉❧t✐♣❧✐❝❛✲

  ∗

  B := R Hom(B, R) çã♦ à ❡sq✉❡r❞❛ ❞♦ s♦✉r❝❡✮✳ ❊♥tã♦ ♦ ❞✉❛❧ à ❡sq✉❡r❞❛ ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ❞❡ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❉❡♠♦♥str❛çã♦✿ ❆♣❧✐❝❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ ✶✳✹✷ ♣❛r❛ ♦ R✲❝♦❛♥❡❧ (B, ∆, ε)✱

  ∗ ∗

  B s : ♦❜t❡♠♦s ✉♠❛ ❡str✉t✉r❛ ❞❡ R✲❛♥❡❧ ❡♠ ✳ ❖ ♠♦r✜s♠♦ ✉♥✐❞❛❞❡ é

  ∗

  R −→

  B, r 7−→ ε(−)r ✱ ♣❛r❛ t♦❞♦ r ∈ R✱ ❡ ♦ ♣r♦❞✉t♦ ❞❡ ❝♦♥✈♦❧✉çã♦ é

  ❞❛❞♦ ♣♦r (ϕ ∗ l ψ)(b) = ψ(b · ϕ(b )) = ψ(t(ϕ(b ))b ),

  (1) (2) (2) (1) ∗

  B ♣❛r❛ q✉❛✐sq✉❡r ψ, ϕ ∈ ❡ b ∈ B✳ ❆❣♦r❛ ❛♣❧✐❝❛♥❞♦ ❛ ✈❡rsã♦ s✐♠étr✐❝❛ ❞❛ ♣r♦♣♦s✐çã♦ ✶✳✹✺ ♣❛r❛ ♦ R✲❛♥❡❧ (B, s)✱ ❝✉❥❛ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦ é ❞❛❞❛ ♣♦r

  ′ ′

  r ◮ b ◭ r = s(r)bs(r ),

  ′

  ∈ R ♣❛r❛ q✉❛✐sq✉❡r r, r ❡ b ∈ B✱ ♦❜t❡♠♦s ✉♠❛ ❡str✉t✉r❛ ❞❡ R✲❝♦❛♥❡❧

  ∗ ∗

  B B ❡♠ ✳ ❉❡st❛ ❢♦r♠❛✱ t❡♠ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦ ❞❛❞❛ ♣♦r

  ′ ′

  (r ⇀ ϕ · r )(b) = ϕ(bs(r))r ,

  ∗ ′

  B ∈ R ♣❛r❛ q✉❛✐sq✉❡r ϕ ∈ ❡ r, r ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱

  ∗ ε · r = ε(−)r = s(r). ∗ ∗

  t : R −→ B ❉❡✜♥✐♠♦s ✱ r 7−→ r ⇀ ε✱ ❞❡ss❛ ❢♦r♠❛

  ∗ t(r)(b) = (r ⇀ ε)(b) = ε(bs(r)).

  

∗ ∗

  ε : B −→ R B ) ❉❡✜♥✐♠♦s t❛♠❜é♠ ❛ ❝♦✉♥✐❞❛❞❡ ✱ ϕ 7−→ ϕ(1 ❡ ♦ ❝♦♣r♦✲

  

∗ ∗ ∗ ∗ n

  ∆ : B −→ B ⊗ R B i ∈ B} ❞✉t♦ ✱ ❞❛❞♦ ❡♠ t❡r♠♦s ❞❛ ❜❛s❡ ❞✉❛❧ {x i =1 ✱

  ∗

  {ρ i ∈ B} ✱ ♣♦r n

  X

  ∗

  ∆(ϕ) = ρ i ⊗ R ϕ(−x i ), i

  =1

  ∗

  B ♣❛r❛ t♦❞♦ ϕ ∈ ✳ ❆❣♦r❛✱ t❡♠♦s q✉❡ ♠♦str❛r ❛s ❝♦♥❞✐çõ❡s r❡st❛♥t❡s

  ∗

  B ♣❛r❛ s❡r ❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ Pr✐♠❡✐r♦✱ ♥♦t❡ q✉❡ ❛ ❡str✉t✉r❛ ❞❡

  ∗ ′ ∗ ′ ∗

  R B l s(r ) ∗ l t(r) ✲❜✐♠ó❞✉❧♦ ❡♠ ✱ r ⇀ ϕ · r ✱ ❝♦✐♥❝✐❞❡ ❝♦♠ ϕ ∗ ✱ ♣❛r❛

  ′ ∗

  ∈ R B q✉❛✐sq✉❡r r, r ❡ ϕ ∈ ✳ ❉❡ ❢❛t♦✱ ❛♥t❡s ♥♦t❡ q✉❡ ∆(s(r)) = ∆(r · 1 B ) = r · ∆(1 B ) = r · 1 B ⊗ 1 B = s(r) ⊗ 1 B ,

  ❞❡ss❛ ❢♦r♠❛

  ∗ ∗

  (ϕ ∗ l t(r))(b) = t(r)(t(ϕ(b ))b )

  (2) (1)

  = ε(t(ϕ(b ))b s(r)

  (2) (1)

  = ε(b s(r) · ϕ(b ))

  (1) (2)

  = ε(b s(r))ϕ(b )

  (1) (2)

  = ϕ(ε(b s(r)) · b )

  (1) (2)

  = ϕ(ε((bs(r)) )(bs(r)) )

  (1) (2)

  = ϕ(bs(r)) = (r ⇀ ϕ)(b). P♦r ♦✉tr♦ ❧❛❞♦✱

  ∗ ∗

  (ϕ ∗ l s(r))(b) = s(r)(t(ϕ(b ))b )

  (2) (1)

  = ε(t(ϕ(b ))b )r

  (2) (1)

  = ε(b )ϕ(b )r

  (1) (2)

  = ϕ(ε(b ) · b )r

  (1) (2)

  = ϕ(b)r = (ϕ · r)(b),

  ∗

  B ♣❛r❛ q✉❛✐sq✉❡r r ∈ R✱ ϕ ∈ ❡ b ∈ B✳ ✈❛♠♦s ✈❡r ❛❣♦r❛ q✉❡ ❛s

  ∗ ∗ ∗ ∗

  s t s t ✐♠❛❣❡♥s ❞❡ ❡ ❝♦♠✉t❛♠✱ q✉❡ é ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❡ q✉❡ é

  ′

  ∈ R ❛♥t✐♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r, r ❡ b ∈ B✱ t❡♠♦s

  ∗ ∗ ′ ∗ ′

  ( t(r) ∗ l s(r ))(b) = ( t(r) · r )(b)

  ∗ ′

  = t(r)(b)r

  ′

  = ε(bs(r))r

  ∗ ′

  = s(r )(bs(r))

  ∗ ′

  = (r ⇀ s(r ))(b)

  ∗ ′ ∗ = ( s(r ) ∗ l t(r))(b).

  ❚❛♠❜é♠ t❡♠♦s

  ∗ ∗ ′ ′ ∗

  ( t(r) ∗ l t(r ))(b) = (r ⇀ t(r))(b)

  ∗ ′

  = t(r)(bs(r ))

  ′

  = ε(bs(r )s(r))

  ′

  = ε(bs(r r))

  ∗ ′

  = t(r r)(b) ❡ ❛✐♥❞❛

  ∗ ∗ ′ ∗ ′

  ( s(r) ∗ l s(r ))(b) = ( s(r) · r )(b)

  ∗ ′

  = s(r)(b)r

  ′

  = ε(b)rr

  ∗ ′ = s(rr )(b). e ∗ ∗ ∗

  B, s, t) P♦rt❛♥t♦✱ t❡♠♦s q✉❡ ( é ✉♠ R ✲ ❛♥❡❧✳ ❆❣♦r❛ ❞❡♥♦t❡ ♦ R✲❛♥❡❧

  (s)

  (B, s) ♣♦r B ❡ ❝♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ❛♣❧✐❝❛çã♦

  

∗ ∗ (s) (s)

  c ( ) : B ⊗ R B → R Hom(B ⊗ R B , R)

  \ ϕ ⊗ ψ 7→ ϕ ⊗ ψ,

  ϕ ⊗ ψ(a ⊗ b) = ψ(a ◭ ϕ(b)) = ψ(as(ϕ(b))) ❡♠ q✉❡ \ ✱ ♣❛r❛ q✉❛✐sq✉❡r

  ∗ ∗ (s) (s)

  ϕ ⊗ ψ ∈ B ⊗ R B ⊗ R B ( ) ❡ a ⊗ b ∈ B ✳ ❈❧❛r♦ q✉❡ c ❡stá ❜❡♠

  ❞❡✜♥✐❞❛✱ ♣♦✐s \

  ϕ ⊗ ψ(r ◮ a ⊗ b) = ψ(s(r)a ◭ ϕ(b)) = ψ(s(r)as(ϕ(b))) = ψ(r · as(ϕ(b))) = rψ(as(ϕ(b))) = r \ ϕ ⊗ ψ(a ⊗ b)

  ✭✷✳✷✷✮ ❡ t❛♠❜é♠

  \ ϕ ⊗ ψ(a ◭ r ⊗ b) = ψ((a ◭ r) ◭ ϕ(b))

  = ψ((a ◭ rϕ(b)) = ψ((a ◭ ϕ(r · b)) = ψ((a ◭ ϕ(r ◮ b)) = \ ϕ ⊗ ψ(a ⊗ r ◮ b).

  ▼♦str❡♠♦s q✉❡ c ( )

  X i

  =1

  (ρ j · F (x j ⊗ x i ))(a ◭ ρ i (b)) = n

  X i,j =1 ρ j (a ◭ ρ i (b))F (x j ⊗ x i )

  = n

  X i,j

  =1

  F (ρ j (a ◭ ρ i (b)) · x j ⊗ x i ) = n

  X i =1 F (a ◭ ρ i (b) ⊗ x i )

  = n

  X i

  =1

  F (a ⊗ ρ i (b) ◮ x i ) = F a ⊗ n

  =1

  = n

  ρ i (b) · x i !

  = F (a ⊗ b), ♣♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s

  ^ \

  ϕ ⊗ ψ = n

  X i,j

  =1

  ρ i ⊗ ρ j · \ ϕ ⊗ ψ(x j ⊗ x i ) = n

  X i,j

  =1

  ρ i ⊗ ρ j · (ψ(x j ◭

  ϕ(x i ))) = n

  X i,j

  =1

  X i,j

  X i,j =1 ρ i ⊗ ρ j · F (x j ⊗ x i ))(a ⊗ b)

  é ✉♠❛ ❜✐❥❡çã♦✳ ❉❡ ❢❛t♦✱ ❞❡✜♥❛ f ( ) : R Hom(B

  é ✐♥✈❡rs❛ ❞❡ c ( )

  (s)

  ⊗ R B

  (s)

  , R) →

  ∗

  B ⊗ R

  ∗

  B F 7→ e

  F , ❡♠ q✉❡ e

  F := P n i,j

  =1 ρ i ⊗ ρ j · F (x j ⊗ x i ).

  ❱❛♠♦s ✈❡r q✉❡ f ( )

  ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r ϕ ⊗ ψ ∈

  ( n

  ∗

  B ⊗ R

  ∗

  B ✱ a ⊗ b ∈ B

  (s)

  ⊗ R B

  (s)

  ❡ F ∈ R Hom(B

  (s)

  ⊗ R B

  (s)

  , R) ✱ t❡♠♦s

  F = be \

  ρ i ⊗ ρ j · (ϕ(x i ) ⇀ ψ)(x j )

  = n

  B ✱ b ∈ B ❡ r ∈ R✱ t❡♠♦s

  (2)

  s(r)(b

  ∗

  s(r) ∗ l ϕ)(b) = ϕ(t(

  ∗

  (

  ∗

  (1)

  ❉❡ ❢❛t♦✱ ❛♥t❡s ♥♦t❡ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r ϕ ∈

  B ✳

  ∗

  B × r R

  ∗

  ∆ ❡stá ❝♦♥t✐❞❛ ❡♠

  ∗

  ))b

  ) = ϕ(t(ε(b

  (1) (b))).

  ∗

  (1)

  ))b

  (2)

  t(r)(b

  ∗

  t(r) ∗ l ϕ)(b) = ϕ(t(

  (

  (2)

  ) = ϕ(t(r)b), t❛♠❜é♠ t❡♠♦s✱

  (1)

  ))b

  (2)

  ) = ϕ(t(r)t(ε(b

  (1)

  )r)b

  ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡ ❛ ✐♠❛❣❡♠ ❞❡

  (as(ϕ

  X i

  X i

  =1

  X i

  = n

  X i =1 ϕ((a ◭ ρ i (b))x i )

  ρ i ⊗ ϕ(−x i )(a ⊗ b) = n

  =1

  ∆(ϕ)(a ⊗ b) = \ n

  X i

  ∗

  \

  ρ i · ϕ(x i ) ⊗ ψ = ϕ ⊗ ψ. ❆❣♦r❛ ♥♦t❡ q✉❡

  =1

  X i

  ρ i ⊗ ϕ(x i ) ⇀ ψ = n

  =1

  ϕ(a(ρ i (b) ◮ x i )) = ϕ a n

  =1

  (2)

  (2)

  ϕ(ab) = ϕ

  B ✱ t❡♠♦s

  ∗

  (b))), ♦✉ s❡❥❛✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ B ❡ ϕ ∈

  (1)

  (as(ϕ

  (b)) = ϕ

  ρ i (b) · x i !!

  (1)

  (a ◭ ϕ

  (2)

  ∆(ϕ)(a ⊗ b) = ϕ

  ∗

  \

  = ϕ(ab), ♣♦r ♦✉tr♦ ❧❛❞♦✱

  )

  = ϕ(t(ε(b

  (2)

  (b

  (2)

  ))))a

  (1)

  s(ψ

  (1)

  (b

  (1)

  ))) = ψ

  (t(ϕ

  s(ϕ

  (2)

  (a

  (2)

  ))a

  (1)

  t(ϕ

  (1)

  (b

  (2)

  (1)

  (2)

  (1)

  (2)

  )) = ψ

  (2)

  (t(ϕ

  (2)

  (a

  (2)

  ◭ ϕ

  (1)

  (b

  )))a

  (a

  (1)

  s(ψ

  (1)

  (b

  (1)

  ))) = ψ

  (2)

  (t(ϕ

  (2)

  ))s(ψ

  (b

  (b

  (2)

  s((ϕ

  (1)

  ∗ l ψ

  (1)

  )(b))) = ψ

  (2)

  (t(ϕ

  (2)

  (a

  ))a

  ))a

  (1)

  s(ψ

  

(1)

  (t(ϕ

  (1)

  (b

  (2)

  ))b

  (1)

  (1)

  (2)

  (1)

  )(b)) = (ϕ

  ))), ♣♦r ♦✉tr♦ ❧❛❞♦✱

  (ϕ

  (2)

  ∗ l ψ

  (2)

  )(a ◭ (ϕ

  (1)

  ∗ l ψ

  

(1)

  (2)

  (a

  ∗ l ψ

  (2)

  )(as((ϕ

  (1)

  ∗ l ψ

  (1)

  )(b))) = ψ

  (2)

  (t(ϕ

  (2)

  (1)

  (1)

  (2)

  (2)

  (ϕ

  (1)

  ⊗

  ∗ R

  t(r) ∗ l ϕ

  (2)

  )(a ⊗ b) =

  ∗

  t(r) ∗ l ϕ

  (a ◭ ϕ

  (t(r)b)) = ϕ(at(r)b),

  (1)

  (b)) = ϕ

  (2)

  (as(ϕ

  (1)

  (b))t(r)) = ϕ

  (2)

  (at(r) ◭ ϕ

  (1)

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ \

  (1)

  ∗

  s(r) ∗ l ϕ

  s(r)))b

  (1)

  ) = ϕ(t(ε(b

  (2)

  ))b

  (1)

  t(r)) = ϕ(bt(r)). P♦rt❛♥t♦✱ t❡♠♦s

  \ (

  ∗

  (1)

  (a ◭ ϕ

  ⊗ R ϕ

  (2)

  )(a ⊗ b) = ϕ

  (2)

  (a ◭ (

  ∗

  s(r) ∗ l ϕ

  (1)

  )(b)) = ϕ

  (2)

  (b)) = ϕ(at(r)b). P♦rt❛♥t♦✱ ❛ ✐♠❛❣❡♠ ❞❡

  ∆ ❡stá ❝♦♥t✐❞❛ ❡♠

  ◭ ψ

  (1)

  (1)

  ∗ l ψ

  (1)

  )(b)), ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ B✳ ❉❡ ❢❛t♦✱ t❡♠♦s

  ϕ ∗ l ψ(ab) = ψ(t(ϕ(a

  (2)

  b

  (2)

  ))a

  b

  

(2)

  (1)

  ) = ψ

  (2)

  (t(ϕ(a

  (2)

  b

  (2)

  ))a

  (1)

  )(a ◭ (ϕ

  ∗ l ψ

  ∗

  ⊗ R (ϕ ∗ l ψ)

  B × r R

  ∗

  B ✳ ❱❛♠♦s ♠♦str❛r

  ❛❣♦r❛ q✉❡

  ∗

  ∆ é ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s✳ ❉❡ ❢❛t♦✱ q✉❡r❡♠♦s ♠♦str❛r q✉❡

  ∗

  ∆(ϕ ∗ l ψ) = (ϕ ∗ l ψ)

  (1)

  (2)

  (2)

  = ϕ

  (1)

  ∗ l ψ

  (1)

  ⊗ R ϕ

  (2)

  ∗ l ψ

  (2)

  , ♦✉ s❡❥❛✱ q✉❡r❡♠♦s ♠♦str❛r q✉❡

  (ϕ ∗ l ψ)(ab) = (ϕ

  )))

  

  = ψ (t(ϕ (a ))a s(( s(ϕ (b )) ∗ l ψ )(b )))

  

(2) (2) (2) (1) (1) (2) (1) (1)

  = ( t(ϕ (b )) ∗ l ψ )(t(ϕ (a ))a s(ψ (b )))

  (1) (2) (2) (2) (2) (1) (1) (1)

  = ψ (t(ϕ (a ))a s(ψ (b ))t(ϕ (b )))

  (2) (2) (2) (1) (1) (1) (1) (2) = ψ (t(ϕ (a ))a t(ϕ (b ))s(ψ (b ))). (2) (2) (2) (1) (1) (2) (1) (1) ∗

  ∆ ❙❡❣✉❡ ❞❛í ❛ ✐❣✉❛❧❞❛❞❡✳ P♦rt❛♥t♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ é ♠♦r✜s♠♦ ❞❡

  ∗ ∗ ∗

  ε

  B, s) á❧❣❡❜r❛s✳ ❆❣♦r❛ ✈❛♠♦s ✈❡r q✉❡ é ❝❛r❛❝t❡r à ❞✐r❡✐t❛ ❡♠ ( ✳ ❉❡

  ∗

  B ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r ϕ, ψ ∈ ✱ t❡♠♦s

  ∗ ∗

  ε(1 B ) = ε(ε) = ε(1 B ) = 1 R , t❛♠❜é♠ t❡♠♦s

  ∗ ∗ ∗ ∗ ∗

  ε( s( ε(ϕ)) ∗ l ψ) = ε( s((ϕ)(1 B )) ∗ l ψ)

  

  = ( s((ϕ)(1 B )) ∗ l ψ)(1 B )

  ∗

  = ψ(t( s(ϕ(1 B ))(1 B ))1 B ) = ψ(t(ε(1 B )ϕ(1 B ))1 B ) = ψ(t(ϕ(1 B ))1 B ) = (ϕ ∗ l ψ)(1 B )

  ∗ = ε(ϕ ∗ l ψ).

  ❈♦♥❝❧✉í♠♦s ❛ss✐♠ ❛ ♣r♦♣♦s✐çã♦✳

  ✷✳✸ ❈♦♥str✉çõ❡s ❞❡ ◆♦✈♦s ❇✐❛❧❣❡❜ró✐❞❡s

  ❱❡r❡♠♦s ♥❡st❛ s❡çã♦ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❜✐❛❧❣❡❜ró✐❞❡s q✉❡ sã♦ ❢♦r✲ ♥❡❝✐❞♦s ♣♦r ❝♦♥str✉çõ❡s ❛ ♣❛rt✐r ❞❡ ♦✉tr♦s ❝♦♥❤❡❝✐❞♦s✳

  ✷✳✸✳✶ ❚✇✐st ❞❡ ❉r✐♥❢❡❧❞

  ❯♠ t✇✐st ❞❡ ❉r✐♥❢❡❧ ❞❡ ✉♠❛ ❜✐á❧❣❡❜r❛ B✱ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k ✱ é ✉♠❛ ❜✐á❧❣❡❜r❛ ❝♦♠ ❛ ♠❡s♠❛ ❡str✉t✉r❛ ❞❡ á❧❣❡❜r❛ ❞❡ B✱ ❡ ❝♦♣r♦✲

  ❞✉t♦ t♦r❝✐❞♦ ♣♦r ✉♠ 2✲❝♦❝✐❝❧♦ ♥♦r♠❛❧✐③❛❞♦ ❡♠ B ✭❝❤❛♠❛❞♦ ❞❡ ❡❧❡♠❡♥t♦ ❉r✐♥❢❡❧✬❞✮✳ ◆❡st❛ s✉❜s❡çã♦ ✈❡r❡♠♦s ♦ ❛♥á❧♦❣♦ t✇✐st ❞❡ ❉r✐♥❢❡❧ ♣❛r❛ ❜✐✲ ❛❧❣❡❜ró✐❞❡s ✈✐st♦ ❡♠ ❬✷✼❪ ❙❡çã♦ ✻✳✸✳ ❚♦rçõ❡s ♠❛✐s ❣❡r❛✐s✱ ❛ q✉❛✐s ♥ã♦ ❝♦rr❡s♣♦♥❞❡♠ ❛ ❡❧❡♠❡♥t♦s ❉r✐♥❢❡❧✬❞✱ sã♦ ❡st✉❞❛❞♦s ❡♠ ❬✸✶❪✳ ❉❡✜♥✐çã♦ ✷✳✶✺ ❙❡❥❛ B ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❯♠ ❡❧❡♠❡♥t♦ J r

  B ✐♥✈❡rtí✈❡❧ ❡♠ B × ✱ é ❝❤❛♠❛❞♦ ✉♠ 2✲❝♦❝✐❝❧♦ ♥♦r♠❛❧✐③❛❞♦ ❡♠ B✱ R q✉❛♥❞♦ s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s✿

  R s(r ))J = J(t(r) ⊗ R s(r )) ∀ r, r ∈ R ′ ′ ′

  ✭✐✮ (t(r) ⊗ ✭❜✐❧✐♥❡❛r✐❞❛❞❡✮❀ R

  1 B )((∆ ⊗ R B)(J)) = (1 B ⊗ R J)((B ⊗ R ∆)(J)) ✭✐✐✮ (J ⊗

  ✭❝♦♥❞✐çã♦ ❞❡ ❝♦❝✐❝❧♦✮❀ R B)(J) = 1 B = (B ⊗ R ε)(J)

  ✭✐✐✐✮ (ε ⊗ ✭♥♦r♠❛❧✐③❛çã♦✮✳ Pr♦♣♦s✐çã♦ ✷✳✶✻ ❙❡❥❛ J ✉♠ 2✲❝♦❝✐❝❧♦ ♥♦r♠❛❧✐③❛❞♦ ❡♠ B✱ ✉♠ ❜✐❛❧❣❡✲ e ❜ró✐❞❡ à ❞✐r❡✐t❛ s♦❜r❡ R✳ ❊♥tã♦ ♦ R ✲❛♥❡❧ (B, s, t)✱ ❛ ❝♦✉♥✐❞❛❞❡ ε ❞❡ B J := J∆(−)J

−1

  ❡ ❛ ❢♦r♠❛ t♦r❝✐❞❛ ∆ ❞♦ ❝♦♣r♦❞✉t♦ ∆ ❡♠ B✱ ❝♦♥st✐t✉✐ J ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛ B ✳ J ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡ ✈❛♠♦s ✈❡r q✉❡ ∆ é ♠♦r✜s♠♦ ❞❡ R✲ ❜✐♠ó❞✉❧♦s✳ ❆♥t❡s✱ ♥♦t❡ q✉❡

  ′ −1 −1 ′ (t(r) ⊗ R s(r ))J = J (t(r) ⊗ R s(r )).

  ✭✷✳✷✸✮

  ′

  ∈ R P♦rt❛♥t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r, r ❡ b ∈ B✱ ♣♦r s✐♠♣❧✐❝✐❞❛❞❡ ✈❛♠♦s i j i ⊗ R J = K j ⊗ R K −1

  ❞❡♥♦t❛r J = J ❡ J ✱ ♦♠✐t✐♥❞♦ ♦ s♦♠❛tór✐♦✱ ❛ss✐♠ t❡♠♦s

  ′ ′ −1

  ∆ J (r · b · r ) = J∆(r · b · r )J

  ′ −1

  = J(r · b ⊗ b · r )J

  (1) (2) ′ −1

  = J(b t(r) ⊗ b s(r ))J

  (1) (2) i ′ j

  = J i b t(r)K j ⊗ J b s(r )K

  

(1) (2)

i j

  = J i b K j t(r) ⊗ J b K s(r )

  (1) (2) ✭♣♦r ✷✳✷✸✮ i j

  = r · J i b K j ⊗ J b K · r

  

(1) (2)

  = r · ∆ J (b) · r . J ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ ❛ ❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞❡ ∆ ✳ ❉❡ ❢❛t♦✱ ❛♥t❡s ♥♦t❡ q✉❡

  −1 −1

  ((∆ ⊗ R B)(J)) = (∆ ⊗ R B)(J ) ❡ ❝♦♠♦

  (1 B ⊗ R J)((B ⊗ R ∆)(J)) = (J ⊗ R

  1 B )((∆ ⊗ R B)(J)), t❡♠♦s✱ ❛♣❧✐❝❛♥❞♦ ♦s ✐♥✈❡rs♦s ❡♠ ❛♠❜♦s ♦s ❧❛❞♦s✱ q✉❡

  −1 −1 −1 −1 ((B ⊗ R ∆)(J ))(1 B ⊗ R J ) = ((∆ ⊗ R B)(J ))(J ⊗ 1 B ).

  P♦rt❛♥t♦✱ ♣❛r❛ t♦❞♦ b ∈ B✱ t❡♠♦s (B ⊗ R ∆ J )∆ J (b)

  −1

  = (B ⊗ R ∆ J )(J∆(b)J ) i j = (B ⊗ R ∆ J )(J i b K j ⊗ J b K )

  (1) (2) i j −1

  = J i b K j ⊗ J∆(J b K )J

  (1) (2) i j −1

  = (1 B ⊗ R J)(J i b K j ⊗ ∆(J b K ))(1 B ⊗ R J )

  (1) (2) −1 −1

  = (1 B ⊗ R J)(B ⊗ R ∆)(J∆J )(1 B ⊗ R J )

  −1 −1

  = (1 B ⊗ R J)(B ⊗ R ∆)(J)((B ⊗ R ∆)∆(b))(B ⊗ R ∆)(J )(1 B ⊗ R J )

  −1 −1

  = (J ⊗ R

  1 B )(∆ ⊗ R B)(J)((∆ ⊗ R B)∆(b))(∆ ⊗ R B)(J )(J ⊗ R

  1 B )

  

−1 −1

  = (J ⊗ R

  1 B )(∆ ⊗ R B)(J∆(b)J )(J ⊗ R

  1 B ) = (∆ J ⊗ R B)∆ J (b). r

  −1

  ∈ B × B ❆❣♦r❛ ♥♦t❡ q✉❡ J R ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ r ∈ R✱ t❡♠♦s j j i

  (s(r)K j ⊗ K )J = s(r)K j J i ⊗ K J = s(r) ⊗ 1 B = 1 B · r ⊗ 1 B = 1 B ⊗ r · 1 B = 1 B ⊗ t(r) j i = K j J i ⊗ t(r)K J j = (K j ⊗ t(r)K )J, j j ✭✷✳✷✹✮

  −1 j ⊗ K = K j ⊗ t(r)K

  ♠✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r J ✱ t❡♠♦s s(r)K ✳ ❆❣♦r❛ ♥♦t❡ q✉❡ i j

  1 B = (ε ⊗ R B)(1 B ⊗ 1 B ) = (ε ⊗ R B)(J i K j ⊗ J K ) i j = ε(J i K j ) · J K i j = J K t(ε(J i K j )) i j = J K t(ε(s(ε(J i ))K j )) i j = J t(ε(J i ))K t(ε(K j )) j = K t(ε(K j ))

  ✭♣♦r ✭✐✐✐✮✮, t❛♠❜é♠ t❡♠♦s j i

  1 B = (B ⊗ R ε)(1 B ⊗ 1 B ) = (B ⊗ ε)(K j J i ⊗ K J ) j i = K j J i · ε(K J ) j i = K j J i · ε(s(ε(K ))J ) i j = K j J i · ε(J s(ε(K )))

  ✭♣♦r ✭✐✮✮ i j

  = K j J i · ε(J )ε(K ) i j = K j J i s(ε(J ))s(ε(K )) j = K j s(ε(K )) j = K j · ε(K ).

  ❙❡❣✉❡ ♣♦rt❛♥t♦✱ ♣❛r❛ t♦❞♦ b ∈ B✱ i j (ε ⊗ R B)∆ J (b) = (ε ⊗ R B)(J i b K j ⊗ J b K )

  (1) (2) i j

  = ε(J i b K j ) · J b K

  (1) (2) i j

  = ε(s(ε(J i ))b K j ) · J b K

  (1) (2) i j

  = ε(b K j ) · J t(ε(J i ))b K

  (1) (2) j

  = ε(b K j ) · b K

  (1) (2) j

  = ε(s(ε(b ))K j ) · b K

  (1) (2) j

  = ε(K j ) · b t(ε(b ))K

  (2) (1) j

  = ε(K j ) · bK j = bK t(ε(K j )) = b, t❛♠❜é♠ t❡♠♦s i j

  (B ⊗ R ε)∆ J (b) = (B ⊗ R ε)(J i b K j ⊗ J b K )

  (1) (2) i j

  = J i b K j · ε(J b K )

  (1) (2) i j

  = J i b K j · ε(s(ε(J b ))K )

  (1) (2) i j

  = J i b K j · ε(t(ε(J b ))K )

  (1) (2) i j

  = J i b s(ε(J b ))K j · ε(K )

  (1) (2) i

  = J i b s(ε(t(ε(J ))b ))

  (1) (2) i

  = J i s(ε(J ))b s(ε(b ))

  (1) (2)

  = b s(ε(b ))

  

(1) (2)

= b. J B B r r

  ➱ ❝❧❛r♦ q✉❡ ❛ ✐♠❛❣❡♠ ❞❡ ∆ ❡stá ❝♦♥t✐❞❛ ❡♠ B × R ✱ ♣♦✐s J ∈ B × R ✳ J ❚❡♠♦s t❛♠❜é♠ q✉❡ ∆ é ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s✱ ♣♦✐s ♣❛r❛ q✉❛✐sq✉❡r

  a, b ∈ B ✱ t❡♠♦s

  −1

  ∆ J (ab) = J∆(ab)J

  −1

  = J∆(a)∆(b)J

  −1 −1

  = J∆(a)J J∆(b)J = ∆ J (a)∆ J (b). J P♦rt❛♥t♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ B é ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳

  ✷✳✸✳✷ ❚✇✐st ♣♦r ✷✲❝♦❝✐❝❧♦

  ❉❡ ♠❛♥❡✐r❛ ❞✉❛❧ à ❝♦♥str✉çã♦ ♥❛ s❡çã♦ ❛♥t❡r✐♦r✱ ♣♦❞❡✲s❡ ❞❡✐①❛r ♦ ❝♦♣r♦❞✉t♦ ✐♥❛❧t❡r❛❞♦ ❡ t♦r❝❡r ♦ ♣r♦❞✉t♦✳ ❙❡❥❛ B ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à e R B e ❡sq✉❡r❞❛✱ ♣♦❞❡♠♦s ♠✉♥✐r ♦ R ✲♠ó❞✉❧♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ B⊗ ✭❝♦♠ r❡s♣❡✐t♦ ❛s ❛çõ❡s à ❞✐r❡✐t❛ ✭à ❡sq✉❡r❞❛✮ ❞❛❞❛s ♣❡❧❛s ♠✉❧t✐♣❧✐❝❛çõ❡s ✭à ❡sq✉❡r❞❛✮ ❞❡ s ❡ t✮ ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ R✲❝♦❛♥❡❧✱ q✉❡ t❡♠ ❡str✉t✉r❛

  ′ ′

  = s(r)t(r )a ⊗ b ❞❡ R✲❜✐♠ó❞✉❧♦ ❞❛❞❛ ♣♦r r · (a ⊗ b) · r ✱ ♣❛r❛ q✉❛✐sq✉❡r

  ′ e

  r, r ∈ R R B ❡ a ⊗ b ∈ B⊗ ✳ ❉❡✜♥✐♠♦s ❝♦♠♦ ❝♦♣r♦❞✉t♦ a ⊗ b 7−→ (a ⊗ b ) ⊗ R (a ⊗ b )

  (1) (1) (2) (2)

  ❡ ❝♦✉♥✐❞❛❞❡ a ⊗ b 7−→ ε(ab). R Hom R (B ⊗ R P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ á❧❣❡❜r❛ ❞❡ ❝♦♥✈♦❧✉çã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ e

  B, R) ✱ ❝♦♠ ♣r♦❞✉t♦ (f ⋄ g)(a ⊗ b) := f (a ⊗ b )g(a ⊗ b ).

  (1) (1) (2) (2)

  ❉❡✜♥✐çã♦ ✷✳✶✼ ❙❡❥❛ B ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛✳ ❯♠ ❡❧❡♠❡♥t♦ R Hom R (B ⊗ R e

  B, R) ✐♥✈❡rtí✈❡❧ σ ∈ é ❝❤❛♠❛❞♦ 2✲❝♦❝✐❝❧♦ ♥♦r♠❛❧✐✲

  ′

  ∈ R ③❛❞♦ ❡♠ B✱ q✉❛♥❞♦ s❛t✐s❢❛③✱ ♣❛r❛ q✉❛✐sq✉❡r r, r ✱ a, b ❡ c ∈ B✱ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿

  ′ ′

  )a, b) = rσ(a, b)r ✭✐✮ σ(s(r)t(r ✭❜✐❧✐♥❡❛r✐❞❛❞❡✮❀

  , c ))b c ) = σ(s(σ(a , b ))a b , c) ✭✐✐✮ σ(a, s(σ(b (1) (1) (2) (2) (1) (1) (2) (2) ✭❝♦♥✲

  ❞✐çã♦ ❞❡ ❝♦❝✐❝❧♦✮❀ B , a) = ε(a) = σ(a, 1 B ) ✭✐✐✐✮ σ(1 ✭♥♦r♠❛❧✐③❛çã♦✮❀

  ✭✐✈✮ σ(a, bs(r)) = σ(a, bt(r))✱ R B e ❡♠ q✉❡ (a, b) ❞❡♥♦t❛ ❡❧❡♠❡♥t♦s ❡♠ B ⊗ ✳ Pr♦♣♦s✐çã♦ ✷✳✶✽ ❙❡❥❛♠ B ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛ ❡ σ ✉♠ 2✲ ❝♦❝✐❝❧♦ ♥♦r♠❛❧✐③❛❞♦ ❡♠ B✱ ❝♦♠ ✐♥✈❡rs❛ eσ✳ ❊♥tã♦ ♦ s♦✉r❝❡ s : R −→ B✱ ♦ t❛r❣❡t t : R −→ B✱ ♦ R✲❝♦❛♥❡❧ (B, ∆, ε) ❡ ♦ ♣r♦❞✉t♦ t♦r❝✐❞♦✱ ❞❡✜♥✐❞♦ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ B✱ ♣♦r a · σ b := s(σ(a , b ))t( , b ))a b ,

  (1) (1) eσ(a (3) (3) (2) (2) σ

  ❝♦♥st✐t✉❡♠ ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛ ❞❡♥♦t❛❞♦ ♣♦r B ✳

  ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r♦ ✈❛♠♦s ✈❡r q✉❡ s ❡ t sã♦ ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❡ q✉❡ s✉❛s ✐♠❛❣❡♥s ❝♦♠✉t❛♠✳ ❉❡ ❢❛t♦✱ ❛♥t❡s ♥♦t❡ q✉❡ ♣❛r❛ t♦❞♦ r ∈ R✱ t❡♠♦s

  (B ⊗ ∆)∆(s(r)) = s(r) ⊗ 1 B ⊗ 1 B = (∆ ⊗ B)∆(s(r)) ❡ (B ⊗ ∆)∆(t(r)) = 1 B ⊗ 1 B ⊗ t(r) = (∆ ⊗ B)∆(t(r)).

  ❚❛♠❜é♠✱ ♣❛r❛ t♦❞♦ b ∈ B✱ t❡♠♦s ε(b) = σ(b , 1 B ) , 1 B )

  

(1) eσ(b (2)

  = , 1 B ))b , 1 B ) eσ(s(σ(b (1) (2) = ))b , 1 B ) eσ(s(ε(b (1) (2) = ) · b , 1 B ) eσ(ε(b (1) (2) = B ). eσ(b, 1 B , b) = ε(b) ∈

  ❆♥❛❧♦❣❛♠❡♥t❡ t❡♠✲s❡ eσ(1 ✳ P♦rt❛♥t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r, r R

  ✱ t❡♠♦s

σ B B

′ ′ s(r) · s(r ) = s(σ(s(r), s(r )))t( , 1 )) eσ(1

  ′

  = s(rσ(1 B , s(r )))t(ε(1 B ))

  

  = s(rε(s(r )))

  ′

  = s(rr ) ✭♣♦r ✷✳✸✮✱ t❛♠❜é♠

  ′ ′

  t(r) · σ t(r ) = s(σ(1 B , 1 B )t( )) eσ(t(r), t(r

  ′

  = t( )) eσ(t(r), t(r

  

  = t(ε(t(r ))r)

  ′

  = t(r r) ✭♣♦r ✷✳✸✮. ❆❣♦r❛ t❡♠♦s

  ′ ′

  t(r) · σ s(r ) = s(σ(1 B , s(r )))t( B )) eσ(t(r), 1

  

  = s(ε(s(r )))t(ε(t(r)))

  

  = s(σ(s(r ), 1 B ))t( B , t(r))) eσ(1

  ′ σ e = s(r ) · σ t(r).

  , s, t) ❙❡❣✉❡ q✉❡ (B é ✉♠ R ✲❛♥❡❧✳ ❉❡♥♦t❡♠♦s ❛ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦ σ ′ ′ σ

  = s(r) · σ t(r ) · σ b ) ⊆ ❡♠ B ♣♦r r ⊲ b ⊳ r ✳ ❱❡r❡♠♦s ❡♥tã♦ q✉❡ ∆(B σ l σ B × B R ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r b ∈ B ❡ r ∈ R✱ t❡♠♦s b · σ t(r) ⊗ b = s(σ(b , 1 B ))t( , t(r)))b ⊗ b

  (1) (2) (1) (3) (2) (4)

  eσ(b

  = s(σ(b , 1 B ))t( , (1 R ⊗ r) · 1 B ))b ⊗ b

  (1) (3) (2) (4)

  eσ(b = s(σ(b , 1 B ))t( · (1 R ⊗ r), 1 B ))b ⊗ b

  (1) (3) (2) (4)

  eσ(b = s(ε(b ))t( t(r), 1 B ))b ⊗ b

  (1) (3) (2) (4)

  eσ(b = s(ε(b ))t( , 1 B ))b ⊗ b s(r)

  (1) (3) (2) (4)

  eσ(b = s(ε(b ))t(ε(b ))b ⊗ b s(r)

  (1) (3) (2) (4)

  = s(ε(b ))b ⊗ b s(r)

  (1) (2) (3)

  = b ⊗ b s(r),

  (1) (2)

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ b ⊗ b · σ s(r) = b ⊗ s(σ(b , s(r)))t( , 1 B ))b

  (1) (2) (1) (2) (4) (3)

  eσ(b = b ⊗ s(σ(b , (r ⊗ 1 B ) · 1 B ))t( , 1 B ))b

  (1) (2) (4) (3)

  eσ(b = b ⊗ s(σ(b · (r ⊗ 1 B ), 1 B ))t( , 1 B ))b

  (1) (2) (4) (3)

  eσ(b = b ⊗ s(σ(b s(r), 1 B ))t(ε(b ))b

  (1) (2) (4) (3)

  = b ⊗ s(ε(b s(r)))t(ε(b ))b

  (1) (2) (4) (3)

  = b t(r) ⊗ s(ε(b ))b

  (1) (2) (3)

  = b t(r) ⊗ b

  (1) (2) = b ⊗ b s(r). (1) (2)

  ❱❛♠♦s ✈❡r q✉❡ ∆ é ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r

  a, b ∈ B ✱ t❡♠♦s

  ∆(a · σ

  b) = ∆(s(σ(a , b ))t( , b ))a b )

  (1) (1) (3) (3) (2) (2)

  eσ(a = σ(a , b ) · ∆(a b ) · , b )

  (1) (1) (2) (2) (3) (3)

  eσ(a = σ(a , b ) · ∆(a )∆(b ) · , b )

  (1) (1) (2) (2) (3) (3)

  eσ(a = σ(a , b ) · a b ⊗ a b · , b )

  (1) (1) (2) (2) (3) (3) (4) (4)

  eσ(a = s(σ(a , b ))a b ⊗ t( , b ))a b ,

  (1) (1) (2) (2) (4) (4) (3) (3)

  eσ(a ♣♦r ♦✉tr♦ ❧❛❞♦✱

  ∆(a) · σ ∆(b) = a · σ b ⊗ a · σ b

  (1) (1) (2) (2)

  = s(σ(a , b ))t( , b ))a b ⊗

  (1)(1) (1)(1) eσ(a (1)(3) (1)(3) (1)(2) (1)(2)

  s(σ(a , b ))t( , b ))a b

  (2)(1) (2)(1) eσ(a (2)(3) (2)(3) (2)(2) (2)(2)

  = s(σ(a , b ))t( , b ))a b ⊗

  (1) (1) (3) (3) (2) (2)

  eσ(a s(σ(a , b ))t( , b ))a b

  (4) (4) (6) (6) (5) (5)

  eσ(a = s(σ(a , b ))a b ⊗

  (1) (1) (2) (2)

  s( , b ))s(σ(a , b ))t( , b ))a b

  (3) (3) (4) (4) (6) (6) (5) (5)

  eσ(a eσ(a

  = s(σ(a , b ))a b ⊗

  (1) (1) (2) (2)

  s( , b )σ(a , b ))t( , b ))a b

  (3) (3) (4) (4) (6) (6) (5) (5)

  eσ(a eσ(a = s(σ(a , b ))a b ⊗

  (1) (1) (2) (2)

  s(ε(a b ))t( , b ))a b

  (3) (3) (5) (5) (4) (4)

  eσ(a = s(σ(a , b ))a b ⊗

  (1) (1) (2) (2)

  t( , b ))s(ε(a b ))a b

  (5) (5) (3) (3) (4) (4)

  eσ(a = s(σ(a , b ))a b ⊗ t( , b ))a b .

  (1) (1) (2) (2) (4) (4) (3) (3)

  eσ(a ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ∆ é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R ❡ b ∈ B✱ t❡♠♦s

  ∆(r ⊲ b) = ∆(s(r) · σ

  b) = ∆(s(r)) · σ ∆(b) = s(r) · σ b ⊗ b

  (1) (2)

  = r ⊲ b ⊗ b

  (1) (2)

  = r ⊲ ∆(b), t❛♠❜é♠ t❡♠♦s ∆(b ⊳ r) = ∆(t(r) · σ

  b) = ∆(t(r)) · σ ∆(b) = b ⊗ t(r) · σ b

  (1) (2)

  = b ⊗ b ⊳ r

  (1) (2) = ∆(b) ⊳ r.

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ε é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛❛✐sq✉❡r r ∈ R ❡ b ∈ B✱ t❡♠♦s ε(r ⊲ b) = ε(s(r) · σ

  b) = ε(s(σ(s(r), b ))t( B , b )b )

  (1) (3) (2)

  eσ(1 = σ(s(r), b )ε(b ) B , b )

  (1) (2) eσ(1 (3)

  = σ(s(r), b ) )), b )

  (1) eσ(s(ε(b (2) (3)

  = σ(s(r), b ) B , s(ε(b ))b )

  (1) eσ(1 (2) (3)

  = rσ(1 B , b ) B , b )

  (1) eσ(1 (2)

  = rε(b), t❛♠❜é♠ t❡♠♦s ε(b ⊳ r) = ε(t(r) · σ

  b)

  = ε(s(σ(1 B , b ))t( ))b )

  (1) (3) (2)

  eσ(t(r), b = σ(1 B , b )ε(b ) )

  (1) (2) (3)

  eσ(t(r), b = ε(b )ε(b ) B , t(r)b )

  (1) (2) (3)

  eσ(1 = ε(s(ε(b ))b ) B , t(r)b )

  (1) (2) (3)

  eσ(1 = ε(b )ε(t(r)b )

  (1) (2)

  = ε(b )ε(b )r

  (1) (2)

  = ε(s(ε(b ))b )r

  

(1) (2)

= ε(b)r.

  ▼♦str❡♠♦s ❛❣♦r❛ ❛ ❝♦✉♥✐t❛❧✐❞❛❞❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ b ∈ B✱ t❡♠♦s b ⊳ ε(b ) = t(ε(b )) · σ b

  (1) (2) (2) (1)

  = s(σ(1 B , b ))t( )), b ))b

  (1) (4) (3) (2)

  eσ(t(ε(b = s(ε(b ))t( B , t(ε(b ))b ))b

  (1) (4) (3) (2)

  eσ(1 = s(ε(b ))t( B , b ))b

  (1) (3) (2)

  eσ(1 = s(ε(b ))t(ε(b ))b

  (1) (3) (2)

  = s(ε(b ))b

  (1) (2)

  = b, t❛♠❜é♠ t❡♠♦s ε(b ) ⊲ b = s(ε(b )) · σ b

  (1) (2) (1) (2)

  = s(σ(s(ε(b )), b ))t( B , b ))b

  (1) (2) (4) (3)

  eσ(1 = s(ε(b )σ(1 B , b ))t(ε(b ))b

  (1) (2) (4) (3)

  = s(ε(b ))s(ε(b ))b

  (1) (2) (3)

  = s(ε(b ))b

  (1) (2) σ = b.

  , ∆, ε) ❙❡❣✉❡ q✉❡ (B é ✉♠ R✲❝♦❛♥❡❧✳ ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡ ε é σ

  , s) ❝❛r❛❝t❡r à ❡sq✉❡r❞❛ ❡♠ (B ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ B✱ t❡♠♦s σ B

  ε(a · s(ε(b))) = ε(s(σ(a , s(ε(b))))t( , 1 ))a )

  (1) eσ(a (3) (2)

  = ε(s(σ(a s(ε(b)), 1 B ))t(ε(a ))a )

  (1) (3) (2)

  = ε(s(σ(a s(ε(b)), 1 B ))a )

  (1) (2)

  = σ(a s(ε(b)), 1 B )ε(a )

  (1) (2)

  = ε(a s(ε(b)))ε(a )

  

(1) (2)

  = ε(a

  , s(ε(b

  c

  (2)

  )), b

  (3)

  c

  (3)

  ) = eσ(a

  (1)

  , b

  (1)

  c

  (1)

  )σ(a

  (2)

  

(2)

  s(ε(b

  c

  (2)

  ))b

  (3)

  c

  (3)

  ) = eσ(a

  (1)

  , b

  (1)

  c

  (1)

  )σ(a

  (2)

  

(2)

  (2)

  (2)

  

(2)

  (3)

  )), b

  (4)

  c

  (4)

  ) = eσ(a

  (1)

  , b

  (1)

  c

  (1)

  )σ(a

  (2)

  s( eσ(b

  , c

  )σ(a

  (4)

  (1)

  c

  (1)

  , b

  (1)

  ) = eσ(a

  c

  (2)

  (4)

  )), b

  (3)

  , c

  (3)

  )σ(b

  , b

  c

  (3)

  c) eσ(a

  , c

  (2)

  ) eσ(a

  (4)

  , b

  (4)

  ) = σ(a

  (1)

  , b

  (1)

  )ε(a

  (2)

  b

  (2)

  

(3)

  b

  , b

  (3)

  ) = σ(a

  (1)

  , b

  (1)

  )ε(a

  (2)

  s(ε(b

  (2)

  c))) eσ(a

  (3)

  , b

  (3)

  (3)

  (3)

  

(2)

  (4)

  ) = ε(abc),

  ✭✷✳✷✻✮ t❛♠❜é♠ t❡♠♦s σ(s(σ(a

  (1)

  , b

  (1)

  ))a

  (2)

  b

  (2)

  , c

  (1)

  ) eσ(t(eσ(a

  (4)

  , b

  ))a

  ) eσ(a

  (1)

  (1)

  , c

  (2)

  b

  (2)

  )σ(a

  , b

  (3)

  (1)

  ) = σ(a

  (2)

  , c

  (3)

  b

  , c

  ))s(σ(b

  (1)

  (3)

  (2)

  ))) eσ(a (3) , b

  (3)

  ) = σ(a

  (1)

  t(ε(b

  (2)

  )), b

  (1)

  )ε(a

  (2)

  ) eσ(a

  (3)

  , b

  ) = σ(a

  

(2)

  (1)

  , t(ε(b

  (2)

  ))b

  (1)

  ) eσ(s(ε(a

  (2)

  ))a

  (3)

  , b

  (3)

  ) = σ(a

  (1)

  , b

  s(ε(b

  )ε(a

  ) eσ(a

  ))a

  s(ε(b)) · ε(a

  (2)

  )) = ε(t(ε(a

  (2)

  ))a

  (1)

  s(ε(b))) = ε(as(ε(b))) = ε(ab),

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ ε(a · σ

  b) = ε(s(σ(a

  (1)

  , b

  (1)

  ))t( eσ(a (3) , b

  (3)

  (2)

  (1)

  b

  , b

  (1)

  ) = σ(a

  (3)

  ) eσ(a (3) , b

  (2)

  

(2)

  b

  )ε(a

  (1)

  , b

  (1)

  ) = σ(a

  (2)

  (1)

  

(2)

  (2)

  (3)

  (1)

  t( eσ(b

  (2)

  , c

  (2)

  )), b

  (1)

  c

  

(1)

  )σ(a

  (2)

  s(σ(b

  (3)

  , c

  )), b

  (4)

  (4)

  c

  (4)

  ) = eσ(a

  (1)

  , b

  (1)

  c

  (1)

  )σ(a

  (2)

  s( eσ(b

  

(2)

  , c

  ) = eσ(a

  c

  , b

  ✭✷✳✷✺✮ ❆♥t❡s ♥♦t❡ q✉❡ eσ(a

  (2)

  ) = ε(ab). ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞♦ ♣r♦❞✉t♦ t♦r❝✐❞♦✳ P❛r❛ t❛♥t♦✱ ✐r❡♠♦s ✉s❛r ♦ ✐t❡♠ ✭✐✐✮ ❞❛ ❉❡✜♥✐çã♦ ✷✳✶✼✳ ❆♥t❡s ♣r❡❝✐s❛♠♦s ❞❡ ✉♠ r❡s✉❧t❛❞♦ ❛♥á❧♦❣♦ ♣❛r❛ eσ✳ ❱❛♠♦s ♠♦str❛r q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ❡ c ∈ B

  ✱ t❡♠♦s eσ(t(eσ(a (2) , b

  (2)

  ))a

  (1)

  b

  (1)

  , c) = eσ(a, t(eσ(b (2) , c

  (2)

  ))b

  (1)

  c

  (1) ).

  (1)

  (4)

  )σ(a

  ))b

  (3)

  , c

  (3)

  , s(σ(b

  (2)

  (1)

  , t( eσ(b

  c

  (1)

  ))b

  (2)

  , c

  (2)

  )

  = σ(a

  (1)

  (1)

  c

  (1)

  ) eσ(t(eσ(a

  (3)

  , b

  

(3)

  ))a

  (2)

  b

  (2)

  , c

  (2)

  ) = ε(a

  b

  (1)

  (2)

  (1)

  b

  (1)

  ) = ε(a

  (2)

  , c

  b

  (1)

  (2)

  ))a

  (3)

  ))) eσ(t(eσ(a (3) , b

  (1)

  t(ε(c

  b

  ❉❡s❡♥✈♦❧✈❡♥❞♦ ♦ ♠❡♠❜r♦ ❡sq✉❡r❞♦✱ t❡♠♦s ε(a

  

(3)

  ) eσ(t(eσ(a

  b

  (2)

  ))a

  (3)

  , b

  

(3)

  (1)

  , c

  c

  (1)

  b

  (1)

  ε(a

  ), ❛❣♦r❛ ✉s❛♥❞♦ ✷✳✷✻ ❡ ✷✳✷✼✱ t❡♠♦s

  (2)

  (2)

  (3) ).

  c

  c

  (3)

  b

  (2)

  )ε(a

  (1)

  (1)

  ) = eσ(a

  ))b

  (2)

  , c

  (2)

  , t( eσ(b

  (1)

  ) eσ(t(eσ(a (3) , b

  ))a

  , c

  (2)

  (1)

  , c) = eσ(s(ε(a

  (1)

  b

  (2)

  ))a

  , b

  (3)

  (3)

  ) eσ(t(eσ(a

  (1)

  , c) = ε(a

  (2)

  ))b

  ))t( eσ(a

  , b

  s(ε(b

  (2)

  (1)

  b

  (2)

  ))a

  

(1)

  ))s(ε(a

  , b

  

(2)

  (3)

  , c) = eσ(t(eσ(a

  (1)

  b

  (2)

  ))a

  (1)

  (2)

  (2)

  (1)

  (2)

  ))a

  

(3)

  ) eσ(t(eσ(a (3) , b

  (1)

  b

  ) = ε(a

  (2)

  (2)

  )), c

  (1)

  s(ε(c

  (2)

  b

  b

  , c) = ε(a

  ))a

  (2)

  (3)

  , b

  (3)

  ) eσ(t(eσ(a

  (1)

  , c) = ε(a

  b

  (1)

  (2)

  ))a

  (3)

  ))) eσ(t(eσ(a (3) , b

  (1)

  t(ε(b

  (4)

  (5)

  (1)

  , b

  t(ε(c))) eσ(a

  (2)

  , b

  (3)

  ) = σ(a

  (1)

  , b

  (1)

  t(ε(c)) eσ(a

  (2)

  , b

  

(2)

  ) = σ(a

  (1)

  (1)

  ))b

  (1)

  ✭✷✳✷✼✮ ❆❣♦r❛ ❞♦ ✐t❡♠ ✭✐✐✮ ❞❛ ❉❡✜♥✐çã♦ ✷✳✶✼✱ t❡♠♦s

  ) = ε(abs(ε(c))) = ε(abc)

  (2)

  , (bs(ε(c)))

  (2)

  ) eσ(a

  , (bs(ε(c)))

  s(ε(c)) eσ(a

  (1)

  = σ(a

  ) ♣♦r (iv)

  

(2)

  , b

  (2)

  (1)

  (2)

  (1)

  (3)

  (2)

  , t(ε(b

  (1)

  ) = σ(a

  (3)

  , b

  ) eσ(a

  (1)

  

(2)

  )ε(a

  (1)

  c)), b

  (2)

  t(ε(b

  c))b

  ) eσ(a

  , t(ε(b

  (1)

  (1)

  ) = σ(a

  (3)

  , b

  (2)

  ) eσ(a

  s(ε(c)))b

  

(2)

  (2)

  , t(ε(b

  (1)

  ) = σ(a

  (3)

  , b

  σ(a, s(σ(b

  , c

  b

  (1)

  (1)

  ))b

  (2)

  , c

  (2)

  , t( eσ(b

  ) = eσ(a

  

(1)

  (5)

  , c

  (5)

  b

  (3)

  ))a

  c

  )σ(s(σ(a

  , b

  (3)

  (4)

  ))a

  (6)

  , b

  (5)

  ) eσ(t(eσ(a

  , c

  (2)

  (4)

  b

  (3)

  ))a

  (3)

  , b

  (6)

  (4)

  (1)

  (1)

  (1)

  , c) ♠✉❧t✐♣❧✐❝❛♥❞♦ ♣❡❧♦ q✉❡ q✉❡r❡♠♦s ♣r♦✈❛r✱ ♣❡❧❛ ❡sq✉❡r❞❛ ❡ ♣❡❧❛ ❞✐r❡✐t❛✱ t❡♠♦s eσ(a

  (2)

  b

  (2)

  ))a

  , b

  (2)

  (1)

  ) = σ(s(σ(a

  (2)

  c

  (2)

  ))b

  , t( eσ(b

  , c

  ) eσ(t(eσ(a

  (3)

  (4)

  c

  (4)

  ))b

  (3)

  , c

  , s(σ(b

  (2)

  (2)

  )σ(a

  (1)

  c

  (1)

  ))b

  , c)

  = eσ(t(eσ(a

  (2)

  (3)

  , b

  (3)

  ) eσ(a

  (1)

  , c

  (2)

  b

  ))a

  (7)

  (1)

  , b

  (1)

  = s(σ(s(σ(a

  (2)

  c

  (5)

  b

  )) t(σ(a

  , b

  ))a

  )) s(σ(a

  b

  (5)

  ))a

  (6)

  ))t( eσ(a (6) , b

  (4)

  , b

  (4)

  (3)

  (7)

  , c

  (8)

  b

  (8)

  ))a

  (9)

  , b

  

(9)

  ) eσ(t(eσ(a

  (5)

  (6)

  c

  , b

  b

  (2)

  ))a

  (3)

  , b

  (3)

  ))t( eσ(a

  (1)

  (1)

  , c

  = s(σ(s(σ(a

  (2)

  )c

  (2)

  · σ b

  (2)

  ))(a

  (3)

  (2)

  (1)

  , b

  (8)

  (6)

  ))t( eσ(a

  (4)

  , b

  (4)

  )) s(σ(a

  (3)

  , c

  b

  )) t( eσ(s(σ(a

  (8)

  ))a

  (9)

  , b

  (9)

  ))t( eσ(a

  (7)

  , b

  (7)

  (5)

  (2)

  (3)

  ))b

  )) s(σ(a

  (7)

  , b

  (6)

  ))t(σ(a

  (4)

  c

  (8)

  (5)

  , b

  , c

  (9)

  , t( eσ(b

  (7)

  )) t( eσ(a

  (3)

  , b

  (2)

  (3)

  (4)

  (2)

  (1)

  (2)

  c

  (2)

  ))b

  (1)

  , c

  (1)

  , s(σ(b

  = s(σ(a

  ))t( eσ(a

  (3)

  c

  (5)

  b

  (4)

  ))a

  (6)

  , b

  (5)

  ))s( eσ(a

  c

  = s(σ(a

  ) eσ(a (2) , b

  (5)

  , c

  ) eσ(a (7) , t( eσ(b (9)

  (7)

  , b

  (6)

  )) ✭♣♦r ✭✐✐✮✲✷✳✶✼✮ t(σ(a

  (3)

  (2)

  (8)

  c

  (2)

  ))b

  (1)

  , c

  (1)

  , s(σ(b

  (1)

  ))b

  c

  (2)

  c

  ))b

  (1)

  , c

  (1)

  , s(σ(b

  (1)

  = s(σ(a

  (3)

  (5)

  (4)

  b

  (4)

  ))a

  (6)

  ))t( eσ(a (5) , b

  (4)

  , b

  (3)

  )) ✭♣♦r ✷✳✷✺✮ s(σ(a

  , c

  · σ b

  (2)

  (1)

  (2)

  , c

  (2)

  )), t( eσ(b

  

(3)

  c

  (3)

  t(ε(b

  ))a

  (1)

  (2)

  = eσ(t(ε(a

  (2) ).

  )ε(a

  (1)

  c

  (1)

  ))b

  ))b

  c

  , c

  c

  , c

  (2)

  )))), t( eσ(b

  (3)

  s(ε(c

  (3)

  = eσ(at(ε(b

  (1) ).

  (1)

  (1) ).

  ))b

  (2)

  , c

  (2)

  )), t( eσ(b

  (3)

  c

  (3)

  = eσ(at(ε(b

  (2)

  

(2)

  ))b

  ))b

  c

  (3)

  b

  (2)

  )ε(a

  (1)

  c

  (1)

  (2)

  = eσ(a

  , t( eσ(b (2) , c

  , c), ❞❡s❡♥✈♦❧✈❡♥❞♦ ♦ ♠❡♠❜r♦ ❞✐r❡✐t♦✱ t❡♠♦s eσ(a (1)

  (1)

  b

  (1)

  ))a

  (2)

  , b

  (3) ).

  (1)

  )), t( eσ(b

  (3)

  (3)

  c

  (3)

  t(ε(b

  (1)

  = eσ(a

  (3) ))).

  c

  s(ε(b

  , t( eσ(b

  (2)

  )ε(a

  (1)

  c

  (1)

  ))b

  (2)

  , c

  (2)

  (2)

  (1)

  (3)

  (1) ).

  

(1)

).

  c

  (1)

  ))b

  (2)

  , c

  (2)

  = eσ(a, t(eσ(b

  c

  b) · σ c = s(σ((a · σ

  (1)

  ))b

  

(2)

  , c

  (2)

  ))b

  (3)

  = eσ(a, t(eσ(t(ε(b

  ❙❡❣✉❡ ♣♦rt❛♥t♦✱ ❛ ✐❣✉❛❧❞❛❞❡✳ P❛r❛ ✜♥❛❧✐③❛r✱ ✈❛♠♦s ✉s❛r ♦ ✐t❡♠ ✭✐✐✮ ❞❛ ❉❡✜♥✐çã♦ ✷✳✶✼ ♣❛r❛ ♠♦str❛r ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞♦ ♣r♦❞✉t♦ t♦r❝✐❞♦✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ❡ c ∈ B✱ t❡♠♦s (a · σ

  b)

  c

  c

  ))t( eσ(a

  (1)

  , c

  (1)

  · σ b

  (1)

  = s(σ(a

  (2)

  (2)

  (1)

  b)

  ))(a · σ

  (3)

  , c

  (3)

  b)

  ))t( eσ((a · σ

  (1)

  , c

  (1) ).

  (1)

  c

  ))b

  , t(ε(c

  (2)

  ))t( eσ(b

  (3)

  = eσ(a, t(ε(b

  (1) ).

  c

  (1)

  (2)

  ))c

  )), c

  (3)

  t(ε(c

  (2)

  )), t( eσ(b

  (3)

  = eσ(at(ε(b

  (1) ).

  (3)

  (2)

  ))b

  (1)

  (3)

  )ε(b

  (2)

  , c

  (2)

  = eσ(a, t(eσ(b

  (1) ).

  c

  ))b

  ))b

  

(2)

  , c

  (2)

  ))t( eσ(b

  (3)

  = eσ(a, t(ε(b

  (1) ).

  c

  (1)

  )) t( eσ(a

  (7)

  (3)

  

(2)

  ))b

  (1)

  , c

  (1)

  , s(σ(b

  (1)

  = s(σ(a

  c

  (2)

  (3)

  b

  (3)

  )a

  (2)

  ))s(ε(a

  (4)

  b

  c

  )) t( eσ(a

  )) t(ε(a

  b

  = s(σ(a

  (3)

  c

  (3)

  b

  (2)

  ))a

  (4)

  (3)

  (4)

  )) t(ε(a

  (4)

  c

  

(5)

  ))b

  (5)

  , c

  (6)

  , t( eσ(b

  (4)

  (4)

  , s(σ(b

  )) t(ε(a

  s(ε(b

  (3)

  )a

  (2)

  ))s(ε(a

  (5)

  b

  (4)

  (4)

  ))b

  c

  

(6)

  ))b

  (5)

  , c

  (7)

  , t( eσ(b

  (5)

  (3)

  (4)

  c

  (2)

  

(5)

  ))b

  (5)

  , c

  (6)

  , t( eσ(b

  (5)

  )) t( eσ(a

  c

  c

  

(2)

  ))b

  (1)

  , c

  (1)

  , s(σ(b

  (1)

  = s(σ(a

  (3)

  (1)

  (1)

  (2)

  (1)

  , t( eσ(b (5) , c

  )) t( eσ(a (4)

  (2)

  c

  

(2)

  ))b

  (1)

  , c

  , s(σ(b

  ))b

  (1)

  = s(σ(a

  (3)

  c

  (3)

  ))b

  (4)

  t(ε(b

  (5)

  

(4)

  )a

  (1)

  (2)

  c

  

(2)

  ))b

  (1)

  , c

  (1)

  , s(σ(b

  = s(σ(a

  c

  (3)

  c

  (3)

  b

  (2)

  )a

  (3)

  )) t(ε(a

  (4)

  (2)

  (3)

  , c

  (5)

  (4)

  s(ε(b

  (3)

  )) t(ε(a

  (4)

  c

  

(5)

  ))b

  , c

  (2)

  (6)

  , t( eσ(b

  (4)

  )) t( eσ(a

  (2)

  c

  

(2)

  ))b

  (1)

  )))a

  b

  )) t(ε(a

  c

  (4)

  c

  

(5)

  ))b

  (5)

  , t( eσ(b (6) , c

  )) t( eσ(a (4)

  (2)

  

(2)

  (3)

  ))b

  (1)

  , c

  (1)

  , s(σ(b

  (1)

  = s(σ(a

  (3)

  c

  )) t( eσ(a

  c

  , t( eσ(b

  , b

  , b

  (5)

  ))t( eσ(a

  (4)

  , b

  (3)

  )σ(a

  (3)

  (2)

  ))a

  )) s( eσ(a

  (7)

  , b

  (6)

  ))t(σ(a

  (4)

  c

  

(8)

  (6)

  (4)

  (5)

  

(2)

  (5)

  , c

  (8)

  , t( eσ(b

  (6)

  )) t( eσ(a

  (2)

  c

  ))b

  b

  (1)

  , c

  (1)

  , s(σ(b

  (1)

  = s(σ(a

  (3)

  c

  (5)

  ))b

  , c

  

(7)

  (7)

  

(4)

  , b

  (3)

  ))s(σ(a

  (3)

  , b

  (2)

  )) s( eσ(a

  , b

  (5)

  (6)

  ))t(σ(a

  (4)

  c

  

(8)

  ))b

  (5)

  , c

  (9)

  ))t( eσ(a

  , b

  (9)

  , c

  , t( eσ(b

  (7)

  )) t( eσ(a

  (2)

  c

  

(2)

  ))b

  (1)

  (1)

  (6)

  , s(σ(b

  (1)

  = s(σ(a

  (3)

  c

  (5)

  b

  (4)

  ))a

  ))b

  c

  

(2)

  ))b

  

(6)

  ))b

  (5)

  , t( eσ(b (7) , c

  )) t( eσ(a (5)

  (2)

  c

  

(2)

  (1)

  (4)

  , c

  (1)

  , s(σ(b

  (1)

  = s(σ(a

  (3)

  c

  (4)

  c

  )) t(ε(a

  (3)

  c

  ))b

  (1)

  , c

  (1)

  , s(σ(b

  (1)

  = s(σ(a

  (3)

  (4)

  (4)

  b

  (3)

  )))a

  

(3)

  t(ε(b

  (2)

  ))s(ε(a

  (5)

  b

  b

  ))a

  (4)

  b

  = s(σ(a

  (3)

  c

  (4)

  b

  (3)

  ))a

  (3)

  (2)

  , s(σ(b

  ))s(ε(a

  (6)

  , b

  (5)

  )σ(a

  (5)

  , b

  (4)

  )) t( eσ(a

  (1)

  (1)

  (3)

  c

  b

  (2)

  ))s(ε(a

  (5)

  b

  (4)

  )) t(ε(a

  (4)

  

(6)

  , c

  ))b

  (5)

  , t( eσ(b (7) , c

  )) t( eσ(a (5)

  (2)

  c

  

(2)

  ))b

  (1)

  )) t( eσ(a

  (3)

  (7)

  (1)

  = s(σ(a

  (4)

  c

  (4)

  b

  (2)

  ))a

  c

  (1)

  (7)

  )) b

  (8)

  , c

  (8)

  ))t( eσ(b

  (6)

  , c

  , s(σ(b

  , c

  )σ(b

  )σ(b

  )) b

  (7)

  , c

  (7)

  ))t( eσ(b

  (5)

  , c

  (5)

  (4)

  

(1)

  , c

  (4)

  , s( eσ(b

  (3)

  )) t( eσ(a

  (2)

  c

  (2)

  ))b

  (6)

  (5)

  c

  ))t( eσ(b

  ))a

  (8)

  c

  (8)

  )) b

  (9)

  , c

  (9)

  (7)

  b

  , c

  (7)

  )σ(b

  (6)

  , c

  (6)

  , s( eσ(b

  (3)

  )) t( eσ(a

  (2)

  (5)

  , c

  (3)

  (5)

  , s( eσ(b

  (3)

  )) t( eσ(a

  (2)

  c

  (2)

  ))b

  c

  c

  (3)

  ))t(ε(b

  

(1)

  , c

  (1)

  , s(σ(b

  (1)

  = s(σ(a

  (5)

  (6)

  (6)

  c

  ))b

  c

  (3)

  b

  (2)

  ))a

  (5)

  c

  (5)

  (4)

  = s(σ(a

  c

  (4)

  ))s(ε(b

  (6)

  , c

  (6)

  , t( eσ(b

  (3)

  (3)

  (1)

  (2)

  ))b

  c

  (3)

  b

  (2)

  ))a

  (4)

  c

  (4)

  (5)

  , s(σ(b

  , t( eσ(b (5) , c

  )) t( eσ(a (3)

  (2)

  c

  (2)

  ))b

  

(1)

  , c

  (1)

  )) t( eσ(a

  c

  ))a

  

(1)

  (4)

  , s(ε(b

  (3)

  )) t( eσ(a

  (2)

  c

  (2)

  ))b

  , c

  (4)

  (1)

  , s(σ(b

  (1)

  = s(σ(a

  (3)

  c

  (3)

  b

  (2)

  c

  ))t( eσ(b

  (2)

  c

  ))b

  

(1)

  , c

  (1)

  , s(σ(b

  (1)

  = s(σ(a

  (3)

  (3)

  (6)

  b

  (2)

  ))a

  (5)

  c

  (5)

  ))b

  (6)

  , c

  (2)

  (2)

  , t( eσ(b

  (3)

  (7)

  , s(σ(b

  (3)

  )) t( eσ(a

  (2)

  c

  (2)

  ))b

  , c

  (7)

  (3)

  ))t( eσ(b

  

(1)

  , c

  (1)

  , s(σ(b

  (1)

  ) = s(σ(a

  , c

  ))t( eσ(b

  · σ c

  , c

  c

  (5)

  ))b

  (6)

  , c

  (6)

  ))t( eσ(b

  (4)

  (4)

  (9)

  s(σ(b

  (2)

  )) a

  (8)

  c

  (8)

  ))b

  (9)

  , c

  (2)

  (2)

  = s(σ(a

  (3)

  ))t( eσ(a

  (1)

  c)

  , (b · σ

  (1)

  , ♣♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s a · σ (b · σ c) = s(σ(a

  (3)

  c

  b

  , (b · σ

  (2)

  )) a

  (4)

  c

  (4)

  ))b

  (5)

  , c

  (5)

  (3)

  c)

  (b

  (1)

  (2)

  ))a

  (3)

  · σ c

  (3)

  , b

  (3)

  ))t( eσ(a

  · σ c

  (3)

  (1)

  , b

  (1)

  = s(σ(a

  (2)

  c)

  (b · σ

  (2)

  ))a

  (5)

  (1)

  )) b

  (2)

  ))t( eσ(b (9) , c

  (7)

  , c

  (7)

  )s(σ(b

  (6)

  , s( eσ(b (6) , c

  )) t( eσ(a (3)

  c

  )) b

  (2)

  )) b

  (3)

  , c

  (3)

  ))t( eσ(b

  (4)

  , c

  (4)

  (9)

  (8)

  

(1)

  (1)

  (4)

  , c

  (4)

  )σ(b

  (3)

  ))t( eσ(b (3) , c

  

(1)

  , c

  , s(σ(b

  c

  (1)

  = s(σ(a

  (5)

  c

  (5)

  b

  (2)

  ))a

  (8)

  ))t(σ(b

  , c

  t(σ(b

  (3)

  (6)

  s( eσ(b

  (3)

  )) t( eσ(a

  (2)

  c

  (2)

  )) b

  , c

  (6)

  (3)

  ))t( eσ(b

  (1)

  , c

  (1)

  )), s(σ(b

  (4)

  , c

  (4)

  , c

  )), s(σ(b

  (1)

  ))a

  , s(σ(b

  (1)

  = s(σ(a

  (5)

  c

  (5)

  b

  (2)

  (8)

  (7)

  c

  (8)

  )) b

  (9)

  , c

  (9)

  ))t( eσ(b

  (7)

  , c

  (3) .

  ❙❡❣✉❡ ♣♦rt❛♥t♦✱ ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞♦ ♣r♦❞✉t♦ t♦r❝✐❞♦✳ ❈♦♠ ✐ss♦✱ ❝♦♥✲ ❝❧✉í♠♦s ❛ ♣r♦♣♦s✐çã♦✳

  ✷✳✸✳✸ ❉✉❛❧✐❞❛❞❡

  ❙❡❥❛ B ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛ ♣r♦❥❡t✐✈♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ❝♦♠♦ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ ✭✈✐❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦ t❛r❣❡t à ❡sq✉❡r❞❛✮✳

  ∗

  := Hom R (B, R) ❊♥tã♦✱ ♣♦❞❡♠♦s ♠✉♥✐r B ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ R✲ e

  ∗ ∗ ∗ ∗

  , s , t ) (r) = ❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✱ ❞❛❞❛ ♣❡❧♦ R ✲❛♥❡❧ (B ✱ ❡♠ q✉❡ s

  ∗ ∗ ∗ ∗

  ε(−t(r)) (r) = rε(−) , ∆ , ε ) ❡ t ✱ ♣❛r❛ t♦❞♦ r ∈ R✱ ♣❡❧♦ R✲❝♦❛♥❡❧ (B ✱ t❛❧ q✉❡ n

  X

  

∗ ∗

  ∆ (ϕ) := ϕ(−x i ) ⊗ ρ i , ∀ ϕ ∈ B , i n n =1 i ∈ B} i ∈ B } := (B, t) ∗ (t) ❡♠ q✉❡ {x i =1 ❡ {ρ i =1 é ❛ ❜❛s❡ ❞✉❛❧ ❞❡ B

  ∗ ∗

  (ϕ) = ϕ(1 B ) ❡ ε ♣❛r❛ t♦❞♦ ϕ ∈ B ❡ ♣❡❧❛ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦

  ′ ∗

  ∈ R ❞❛❞❛✱ ♣❛r❛ q✉❛✐sq✉❡r r, r ✱ ϕ ∈ B ❡ b ∈ B✱ ♣♦r

  ′ ′ (r · ϕ ↼ r )(b) = rϕ(bt(r )).

  ❖ ♣r♦❞✉t♦ ❞❡ ❝♦♥✈♦❧✉çã♦ é ❞❛❞♦ ♣♦r (ϕ ∗ r ψ)(b) = ψ(ϕ(b ) · b ) = ψ(s(ϕ(b ))b ),

  (1) (2) (1) (2) ∗

  ♣❛r❛ q✉❛✐sq✉❡r ϕ, ψ ∈ B ❡ b ∈ B✳ ❖s ❞❡t❛❧❤❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ✭❬✶✺❪✱ Pr♦♣♦s✐çã♦ ✷✳✺✮✳ ❆s ❝♦♥str✉çõ❡s ♥❛s s❡çõ❡s ✷✳✸✳✶ ❡ ✷✳✸✳✷ sã♦ ❞✉❛✐s✱ ♥♦ s❡♥t✐❞♦ ❞❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✳ Pr♦♣♦s✐çã♦ ✷✳✶✾ ❙❡❥❛ B ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛ ♣r♦❥❡t✐✈♦ ✜✲ ♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ❝♦♠♦ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✱ ✭✈✐❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦

  ∗

  t❛r❣❡t à ❡sq✉❡r❞❛✮✳ ❈♦♥s✐❞❡r❡ ♦ ❞✉❛❧ à ❞✐r❡✐t❛ B ❝♦♠ ❡str✉t✉r❛ ❞❡ R✲ i r i ⊗ R ϕ ∈ B × B ∗ ∗ ❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❊♥tã♦ ✉♠ ❡❧❡♠❡♥t♦ J = ϕ R

  

  é ✉♠ 2✲❝♦❝✐❝❧♦ ♥♦r♠❛❧✐③❛❞♦ ❡♠ B s❡✱ ❡ s♦♠❡♥t❡ s❡✱ n

  X i e ′ ′ σ J : B ⊗ R B −→ R, σ J (b, b ) := ϕ i (bt(ϕ (b ))) i

  =1

  é ✉♠ 2✲❝♦❝✐❝❧♦ ♥♦r♠❛❧✐③❛❞♦ ❡♠ B✳

  ∗

  ❉❡♠♦♥str❛çã♦✿ (⇒) ❖ ❢❛t♦ ❞❡ J s❡r 2✲❝♦❝✐❝❧♦ ♥♦r♠❛❧✐③❛❞♦ ❡♠ B

  ′

  ∈ R ♥♦s ❞✐③ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r r, r ✱ t❡♠♦s i i

  ∗ ∗ ′ ∗ ∗ ′

  (r)∗ r ϕ i ⊗ R s (r )∗ r ϕ = ϕ i ∗ r t (r)⊗ R ϕ ∗ r s (r ) ✭❛✮ t

  ✭❜✐❧✐♥❡❛r✐❞❛❞❡✮❀

  ✭❜✮ (J ⊗ R ε)((∆

  (1)

  (s

  ) = ϕ(s(r)b), t❛♠❜é♠ t❡♠♦s

  (2)

  ))b

  (1)

  ) = ϕ(s(r)s(ε(b

  (2)

  ) · b

  ) = ϕ(rε(b

  (r) ∗ r ϕ)(b) = ϕ(s

  (2)

  ) · b

  (1)

  (r)(b

  ∗

  (r) ∗ r ϕ)(b) = ϕ(t

  ∗

  ✱ t❡♠♦s (t

  ∗

  ∗

  ✱ b ∈ B ❡ ϕ ∈ B

  ) ⊲ b

  σ J (at(r), b) = ϕ i (at(r)t(ϕ i (b))) = ϕ i (at(ϕ i (b)r)) = ϕ i (at(ϕ i (t(r)b))) = σ J (a, t(r)b), t❛♠❜é♠ t❡♠♦s

  B ❡ r ∈ R✱ t❡♠♦s

  s(r)) = ϕ(bs(r)). ❱❛♠♦s ✈❡r ❛❣♦r❛ q✉❡ σ J ❡stá ❜❡♠ ❞❡✜♥✐❞♦✱ ♦✉ s❡❥❛✱ q✉❡ é R e ✲❜❛❧❛♥❝❡❛❞♦ ❡ R✲❜✐❧✐♥❡❛r✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r (a, b) ∈ B ⊗ R e

  (2)

  ))b

  (1)

  s(r)) = ϕ(s(ε(b

  (2)

  (1)

  (r)(b

  ) = ϕ(ε(b

  (2)

  t(r)) ⊲ b

  (1)

  ) = ϕ(ε(b

  (2)

  ) ⊲ b

  (1)

  ∗

  ✭✐✈✮ σ J (a, s(r)) = σ J (a, t(r)) ✳ ▼♦str❡♠♦s ❛❣♦r❛ ❞✉❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ✈ã♦ ♥♦s ❛❥✉❞❛r✳ P❛r❛ q✉❛✐sq✉❡r r ∈ R

  ∗

  ⊗ R B

  ′

  ❚❡♠♦s q✉❡ ♠♦str❛r✱ ♣❛r❛ q✉❛✐sq✉❡r r, r

  )(J) ✭♥♦r♠❛❧✐③❛çã♦✮✳

  ∗

  ⊗ R ε

  ∗

  )(J) = (B

  ∗

  ∗

  ✭✐✮ σ J (s(r)t(r

  ❞❡ ❝♦❝✐❝❧♦✮❀ ✭❝✮ (ε

  )(J)) ✭❝♦♥❞✐çã♦

  ∗

  ⊗ R ∆

  ∗

  )(J)) = (ε ⊗ R J)((B

  ∗

  ⊗ R B

  ∈ R ❡ a, b ∈ B✱

  ′

  ❞✐çã♦ ❞❡ ❝♦❝✐❝❧♦✮❀ ✭✐✐✐✮ σ J (1 B , a) = ε(a) = σ J (a, 1 B ) ✭♥♦r♠❛❧✐③❛çã♦✮❀

  ) = σ J (s(σ J (a

  , c) ✭❝♦♥✲

  (2)

  b

  (2)

  ))a

  (1)

  , b

  (1)

  (2)

  )a, b) = rσ J (a, b)r

  c

  (2)

  ))b

  (1)

  , c

  (1)

  ✭❜✐❧✐♥❡❛r✐❞❛❞❡✮ ✭✐✐✮ σ J (a, s(σ J (b

  ′

  σ J (as(r), b) = ϕ i (as(r)t(ϕ i (b)))

  = ϕ i (at(ϕ i (b))s(r)) = (s

  ε

  ∗

  (r) ∗ r ϕ i )(at(ϕ i (b))) = (ϕ i ∗ r t

  ∗

  (r))(at(ϕ i (b))) ✭♣♦r ✭❛✮✮

  = (r · ϕ i )(at(ϕ i (b))) = rϕ i (at(ϕ i (b))) = rσ J (a, b).

  ❆❣♦r❛ ♥♦t❡ q✉❡ ❛ ❝♦♥❞✐çã♦ ❞❡ ♥♦r♠❛❧✐③❛çã♦ ❞❡ σ J ❞❡❝♦rr❡ ❞❛ ♥♦r♠❛❧✐✲ ③❛çã♦ ❞❡ J✳ ❉❡ ❢❛t♦✱ ❛ ♥♦r♠❛❧✐③❛çã♦ ❞❡ J ♥♦s ❞✐③ q✉❡

  ∗

  = ϕ i (at(ϕ i (b)) ⊳ r) = ϕ i (at(ϕ i (b)))r = σ J (a, b)r, t❛♠❜é♠

  (ϕ i ) · ϕ i = 1 B = ε = ϕ i ↼ ε

  ∗

  (ϕ i ), ♣♦rt❛♥t♦✱ s❡❣✉❡ q✉❡

  ε(b) = (ε

  ∗

  (ϕ i ) · ϕ i )(b) = ε

  ∗

  σ J (s(r)a, b) = ϕ i (s(r)at(ϕ i (b))) = (t

  ❡ r ∈ R✱ t❡♠♦s σ J (t(r)a, b) = ϕ i (t(r)at(ϕ i (b)))

  ∗

  ✮ = ϕ i (at(ϕ i (s(rε(b

  (r) ∗ r ϕ i )(at(ϕ i (b))) = ϕ i (at((t

  ∗

  (r) ∗ r ϕ i )(b))) ✭♣♦✐s J ∈ B

  ∗

  × r R B

  ∗

  

(1)

  ▼♦str❡♠♦s ❛❣♦r❛✱ ❛ ❝♦♥❞✐çã♦ ❞❡ ❜✐❧✐♥❡❛r✐❞❛❞❡ ♣❛r❛ σ J ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r (a, b) ∈ B ⊗ R e B

  ))b

  (2)

  ))) = ϕ i (at(ϕ i (s(r)s(ε(b

  (1)

  ))b

  (2)

  ))) = ϕ i (at(ϕ i (s(r)b))) = σ J (a, s(r)b).

  (ϕ i )ϕ i (b) = ϕ i (1 B )ϕ i (b) = ϕ i (t(ϕ i (b))) = σ J (1 B , b),

  ❡

  ∗ i

  ε(b) = (ϕ i ↼ ε (ϕ ))(b) i

  ∗

  = ϕ i (bt(ε (ϕ ))) i = ϕ i (bt(ϕ (1 B ))) J = σ J (b, 1 B ).

  ❆ ❝♦♥❞✐çã♦ ❞❡ ❝♦❝✐❝❧♦ ❞❡ σ ❞❡❝♦rr❡ t❛♠❜é♠ ❞❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦❝✐❝❧♦ ❞❡ J

  ✳ P❛r❛ ♠♦str❛r♠♦s ✐st♦✱ ✈❛♠♦s ♣r❡❝✐s❛r ❞❛s s❡❣✉✐♥t❡s ❜✐❥❡çõ❡s

  

∗ ∗ (t) (t)

  c ( ) : B ⊗ R B −→ Hom R (B ⊗ R B , R)

  \ ϕ ⊗ ψ 7−→ ϕ ⊗ ψ,

  (t) (t)

  ϕ ⊗ ψ(a ⊗ b) = ϕ(bt(ψ(a))) ⊗ R B t❛❧ q✉❡ \ ✱ ♣❛r❛ t♦❞♦ a ⊗ b ∈ B ✱ ❡

  

∗ ∗ ∗ (t) (t) (t)

  f ( ) : B ⊗ R B ⊗ R B −→ Hom R (B ⊗ R B ⊗ R B , R)

  ^ ϕ ⊗ ψ ⊗ η 7−→ ϕ ⊗ ψ ⊗ η,

  ϕ ⊗ ψ ⊗ η(a ⊗ b ⊗ c) = ϕ(ct(ψ(bt(η(a))))) t❛❧ q✉❡ ^ ✱ ♣❛r❛ t♦❞♦ a ⊗ b ⊗ c ∈

  ∗ ∗ ∗

  B ⊗ R B ⊗ R B ✳ ❊ss❛s ❜✐❥❡çõ❡s ♣♦❞❡♠ s❡r❡♠ ♠♦str❛❞❛s ❞❡ ♠❛♥❡✐r❛

  ❛♥á❧♦❣❛ ❛♦ q✉❡ ❢♦✐ ❢❡✐t♦ ♥❛ ♣r♦♣♦s✐çã♦ ✷✳✶✹✳ ❯s❛♥❞♦ ❛ ♣r✐♠❡✐r❛ ❜✐❥❡çã♦ t❡♠♦s n \ !

  X \

  ∗

  ∆ (ϕ)(b ⊗ a) = ϕ(−x j ) ⊗ ρ j (b ⊗ a) n i =1

  X \ j j

  = (ϕ(−x ) ⊗ ρ )(b ⊗ a) i

  =1 n

  X = ϕ(a t(ρ j (b))x j ) i

  =1

  = ϕ(ab), ♣♦r ♦✉tr♦ ❧❛❞♦✱

  \

  ∗ \

  ∆ (ϕ)(b ⊗ a) = ϕ ⊗ ϕ (b ⊗ a)

  (1) (2)

  = ϕ (a t(ϕ (b))),

  (1) (2) ∗

  ♦✉ s❡❥❛✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ B ❡ ϕ ∈ B ✱ t❡♠♦s ϕ(ab) = ϕ (b t(ϕ (a))).

  (1) (2) ✭✷✳✷✽✮

  ❆❣♦r❛ ❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦❝✐❝❧♦ ♣❛r❛ J ♥♦s ❞✐③ q✉❡ j i i j i ϕ j ∗ r ϕ ⊗ ϕ ∗ r ϕ ⊗ ϕ = ϕ i ⊗ ϕ j ∗ ϕ ⊗ ϕ ∗ ϕ . i (1) i (2)

  (1) (2) ∗ ∗ ∗

  ⊗ R B ⊗ R B ❆♣❧✐❝❛♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ ❡♠ a ⊗ b ⊗ c ∈ B ✱ t❡♠♦s

  ^ j i (ϕ j ∗ r ϕ i ⊗ ϕ ∗ r ϕ i ⊗ ϕ )(a ⊗ b ⊗ c)

  (1) (2) j i

  = (ϕ j ∗ r ϕ i )(c t(ϕ ∗ r ϕ i (bt(ϕ (a))))

  (1) (2) j i

  = (ϕ j ∗ r ϕ i )(c t(ϕ i (ϕ (b ) · b t(ϕ (a))))) j r (1) (2) (1) (2) j i = (ϕ ∗ ϕ i )(c t(ϕ i (s(ϕ (b ))b t(ϕ (a)))))

  (1) (2) (1) (2) j i

  = ϕ i (ϕ j (c ) · c t(ϕ i (s(ϕ (b ))b t(ϕ (a)))))

  (1) (1) (2) (2) (1) (2) j i

  = ϕ i (s(ϕ j (c ))c t(ϕ i (s(ϕ (b ))b t(ϕ (a))))),

  (1) (1) (2) (2) (1) (2) ✭✷✳✷✾✮

  ❡ ^ i i j

  (ϕ i ⊗ ϕ j ∗ ϕ ⊗ ϕ ∗ ϕ )(a ⊗ b ⊗ c)

  (1) (2) i j i

  = ϕ i (c t((ϕ j ∗ r ϕ )(b t((ϕ ∗ r ϕ )(b))))) (1) (2) i i j = ϕ i (c t((ϕ j ∗ r ϕ )(b t(ϕ (ϕ (a ) · a )))))

  (1) (2) i i j (1) (2)

  = ϕ i (c t(ϕ (ϕ j (b ) · b t(ϕ (ϕ (a ) · a )))))

  (1) (2) (1) (2) i i j (1) (2) = ϕ i (c t(ϕ (s(ϕ j (b ))b t(ϕ (s(ϕ (a ))a ))))).

  (1) (2) (1) (2) ✭✷✳✸✵✮ (1) (2)

  ▲❡♠❜r❡✱ q✉❡r❡♠♦s ♠♦str❛r q✉❡ σ J (a, s(σ J (b , c ))b c ) = σ J (s(σ J (a , b ))a b , c).

  (1) (1) (2) (2) (1) (1) (2) (2)

  ❉❡ ❢❛t♦✱ t❡♠♦s σ J (a, s(σ J (b , c ))b c ) i (1) (1) (2) (2)

  = ϕ i (a t(ϕ (s(σ J (b , c ))b c ))) i j (1) (1) (2) (2) = ϕ i (a t(ϕ (s(ϕ j (b t(ϕ (c ))))b c )))

i j

(1) (1) (2) (2)

  = ϕ i (a t(ϕ (s(ϕ j (b ))b s(ϕ (c ))c ))),

  (1) (2) (1) (2)

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ σ J (s(σ J (a , b ))a b , c)

  (1) (1) (2) (2) i

  = ϕ i (s(σ J (a , b ))a b t(ϕ (c)))

  (1) (1) (2) (2) j i

  = ϕ i (s(ϕ j (a t(ϕ (b ))))a b t(ϕ (c)))

  (1) (1) (2) (2) j i

  = ϕ i (s(ϕ j (a ))a s(ϕ (b ))b t(ϕ (c)))

  (1) (2) (1) (2) j i

  = ϕ i (s(ϕ j (a ))a t(ϕ i (s(ϕ (b ))b t(ϕ (c)))))

  

(1) (1) (2) (2) (1) (2) ✭♣♦r ✷✳✷✽✮

i i j

  = ϕ i (at(ϕ (s(ϕ j (b ))b t(ϕ (s(ϕ (c ))c ))))) ✭♣♦r ✷✳✷✾ ❡ ✷✳✸✵✮

  (1) (1) (2) (2) (1) (2) i j = ϕ i (a t(ϕ (s(ϕ j (b ))b s(ϕ (c ))c ))).

  (1) (2) (1) (2) J (b, s(r)) = σ J (b, t(r))

  ❙❡❣✉❡ ❛ ✐❣✉❛❧❞❛❞❡✳ ❋❛❧t❛ ♠♦str❛r♠♦s q✉❡ σ ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r b ∈ B ❡ r ∈ R✱ t❡♠♦s i

  σ J (b, s(r)) = ϕ i (b t(ϕ (s(r)))) i

  ∗

  = ϕ i (b t((ϕ ∗ r s (r))(1 B ))) i = ϕ i (b t((ϕ ↼ r)(1 B ))) i = ϕ i (b t(ϕ (1 B t(r)))) i = ϕ i (b t(ϕ (t(r)))) = σ J (b, t(r)). j −1

  −1 1

  = ψ j ⊗ R ψ = σ J ❆❣♦r❛ ❞❡♥♦t❡ J ❡ ♠♦str❡♠♦s q✉❡ σ ✳ ❉❡ ❢❛t♦✱ i ∗ r ψ j ⊗ R ϕ ∗ r ψ = ε ⊗ R ε i j J ♥♦t❡ q✉❡ ϕ ✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❛r❛ q✉❛✐sq✉❡r

  a, b ∈ B ✱ t❡♠♦s

  \ \ i j (ϕ i ∗ r ψ j ⊗ R ϕ ∗ r ψ )(a ⊗ b) = (ε ⊗ R ε)(a ⊗ b)

  = ε(b t(ε(a))) = ε(b s(ε(a))) = ε(ba),

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ \ i j

  (ϕ i ∗ r ψ j ⊗ R ϕ ∗ r ψ )(a ⊗ b) = (ϕ i ∗ r ψ j )(b t((ϕ i ∗ r ψ j )(a))) j i = (ϕ i ∗ r ψ j )(b t(ψ (s(ϕ (a ))a ))) j i (1) (2)

  = ψ j (s(ϕ i (b ))b t(ψ (s(ϕ (a ))a )))

  (1) (2) (1) (2) i j

  = ψ j (s(ϕ i (b ))b t((t (ϕ (a ) ∗ r ψ )(a ))))

  (1) (2) (1) (2) ∗ i j

  = (s (ϕ (a )) ∗ r ψ j )(s(ϕ i (b ))b t(ψ (a )))

  (1) (1) (2) (2) j i

  = ψ j (s(ϕ i (b ))b t(ψ (a ))s(ϕ (a )))

  (1) (2) (2) (1) i j

  = ψ j (s(ϕ i (b ))b s(ϕ (a ))t(ψ (a )))

  (1) (2) (1) (2) i j

  = ψ j (s(ϕ i (b t(ϕ (a ))))b t(ψ (a )))

  (1) (1) (2) (2) j

  = ψ j (s(σ J (b , a ))b t(ψ (a ))) 1

(1) (1) (2) (2)

= σ (s(σ J (b , a ))b , a ) J (1) (1) (2) (2)

  = σ J (b

  )))) = σ J (s(r)b, a s(r

  ))(a ⊗ b) = (ϕ i ∗ r t

  ′

  (r

  ∗

  (r) ⊗ R ϕ i ∗ r s

  ∗

  \ (ϕ i ∗ r t

  )), ♣♦r ♦✉tr♦ ❧❛❞♦✱

  ′

  )) = rσ J (b, a t(r

  ′

  

  (r))(b t((ϕ i ∗ r s

  )))) = ϕ i (s(r)b t(ϕ i (a s(r

  ′

  (r) ∗ r ϕ i )(b t(ϕ i (a s(r

  ∗

  )))) = (t

  ′

  s(r

  (2)

  )a

  (1)

  (r) ∗ r ϕ i )(b t(ϕ i (s(ε(a

  ∗

  ∗

  ))) = (t

  

  ∗

  (r) ∗ r ϕ i ⊗ R ϕ i )(a ⊗ b) = (s

  ∗

  \ (s

  ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R ❡ a ⊗ b ∈ B ⊗ R e B ✱ t❡♠♦s

  

  × r R B

  ∗

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ J ∈ B

  ′ )).

  )))) = rσ J (b, a t(r

  )))) = r ϕ i (b t(ϕ i (a t(r

  (r

  ′

  )))) = (r · ϕ i )(b t(ϕ i (a t(r

  ′

  (r))(b t(ϕ i (a t(r

  ∗

  )(a))) = (ϕ i ∗ r t

  ′

  (r))(b t((ϕ i ↼ r

  ∗

  ))(a))) = (ϕ i ∗ r t

  ′

  ∗

  (2)

  (1)

  (2)

  ∈ R ✱ t❡♠♦s

  ′

  ✳ ▼♦str❡♠♦s ❛❣♦r❛ ❛ ❝♦♥❞✐çã♦ ❞❡ ❜✐❧✐♥❡❛r✐❞❛❞❡ ♣❛r❛ J✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ B ❡ r, r

  ∗

  × r R B

  ∗

  ❡ ♥♦r♠❛❧✐③❛çã♦✱ ✈❛❧❡♠ ♣❛r❛ σ J s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ✈❛❧❡♠ ♣❛r❛ J✳ ❆❧é♠ ❞✐ss♦✱ σ J é ✐♥✈❡rsí✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ J é ✐♥✈❡rsí✈❡❧✳ P♦rt❛♥t♦✱ r❡st❛ ♠♦str❛r♠♦s ❛ ❝♦♥❞✐çã♦ ❞❡ ❜✐❧✐♥❡❛r✐❞❛❞❡ ♣❛r❛ J ❡ q✉❡ J ∈ B

  (⇐) ❉♦ q✉❡ ✜③❡♠♦s ❛❝✐♠❛✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦❝✐❝❧♦

  ) = ε(ba) ✳

  (2)

  , a

  )σ J (b

  ∗

  (1)

  , a

  (1)

  (b

  ), ❛♥❛❧♦❣❛♠❡♥t❡✱ ♠♦str❛✲s❡ q✉❡ σ J 1

  (2)

  , a

  

(2)

  )σ J 1 (b

  (1)

  , a

  \ (t

  (r) ∗ r ϕ i ⊗ R s

  ))a

  (r

  ′

  t(r

  (1)

  (r) ∗ r ϕ i )(b t(ϕ i (s(ε(a

  ∗

  ))) = (t

  (2)

  ))a

  (1)

  )(a

  ′

  ∗

  ∗

  (r) ∗ r ϕ i )(b t(ϕ i (s(s

  ∗

  ) ∗ r ϕ i )(a))) = (t

  ′

  (r

  

  (r) ∗ r ϕ i )(b t((s

  ∗

  ) ∗ r ϕ i )(a ⊗ b) = (t

  ′

  (r

  (r) ∗ r ϕ i )(b t(ϕ i (a))) = ϕ i (b t(ϕ i (a))s(r)) = ϕ i (b s(r)t(ϕ i (a)))

  = σ J (bs(r), a) = σ J (b, s(r)a) i = ϕ i (b t(ϕ (s(r)a))) i

  ∗

  = ϕ i (b t((s (r) ∗ r ϕ )(a))) \ i

  ∗ = (ϕ i ⊗ R t (r) ∗ r ϕ )(a ⊗ b). ∗ i ∗ i ∗ r ∗

  (r)∗ r ϕ i ⊗ R ϕ = ϕ i ⊗ R t (r)∗ r ϕ × B ❙❡❣✉❡ q✉❡ s ✳ P♦rt❛♥t♦✱ J ∈ B R ✳ ❈♦♥❝❧✉í♠♦s ❛ss✐♠✱ ❛ ❞❡♠♦♥str❛çã♦✳

  ✷✳✸✳✹ ❇✐❛❧❣❡❜ró✐❞❡ ❞❡ ❈♦♥♥❡s✲▼♦s❝♦✈✐❝✐

  ❖ ❜✐❛❧❣❡❜ró✐❞❡ ❛ s❡❣✉✐r ❢♦✐ ❝♦♥str✉í❞♦ ♥❛ t❡♦r✐❛ ❞❡ ❢♦❧❤❡❛çõ❡s ♥♦ ❝♦♥t❡①t♦ ❞♦ ❝á❧❝✉❧♦ ❞♦ í♥❞✐❝❡ tr❛♥s✈❡rs❛❧ ❞❡ ❢♦❧❤❡❛çõ❡s ❡ ♣♦❞❡ s❡r ❡♥✲ ❝♦♥tr❛❞♦ ❡♠ ❬✶✵❪✳ Pr♦♣♦s✐çã♦ ✷✳✷✵ ❈♦♥s✐❞❡r❡ H ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k✱ ❡ s❡❥❛ R ✉♠ H✲♠ó❞✉❧♦ á❧❣❡❜r❛ à ❡sq✉❡r❞❛✳ ❈♦♥s✐❞❡r❡ ♦ k k H ⊗ k R

  ✲♠ó❞✉❧♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ B := R ⊗ ❡q✉✐♣❛❞♦ ❝♦♠ ❛ ♠✉❧t✐✲ ♣❧✐❝❛çã♦ ❛ss♦❝✐❛t✐✈❛

  ′ ′ ′ ′

  (r ⊗ h ⊗ s)(r ⊗ k ⊗ s ) := r(h ⊲ r ) ⊗ h k ⊗ (h ⊲ s )s,

  (1) (2) (3) ′ ′

  ⊗ k ⊗ s ∈ B ♣❛r❛ q✉❛✐sq✉❡r r ⊗ h ⊗ s, r ✳ ❊♥tã♦✱ P♦❞❡♠♦s ♠✉♥✐r B ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛✱ ❝♦♠ s♦✉r❝❡ ❡ t❛r❣❡t H ⊗ 1 R ❞❛❞♦s✱ ♣❛r❛ t♦❞♦ r ∈ R✱ ♣♦r s : R −→ B, r 7−→ r ⊗ 1 ❡ t : R −→ B, r 7−→ 1 R ⊗ 1 H ⊗ r

  ✳ ❉❡✜♥✐♠♦s t❛♠❜é♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ R

  ✲❝♦❛♥❡❧ (B, ∆, ε)✱ ♣♦r ∆(r ⊗ h ⊗ s) = (r ⊗ h ⊗ 1 R ) ⊗ R (1 R ⊗ h ⊗ s)

  (1) (2)

  ❡ ε(r ⊗ h ⊗ s) = rε(h)s,

  ♣❛r❛ t♦❞♦ r ⊗ h ⊗ s ∈ B✳ ❉❡♠♦♥str❛çã♦✿ ❱❡r❡♠♦s ♣r✐♠❡✐r♦ q✉❡ s♦✉r❝❡ ❡ t❛r❣❡t ❝♦♠✉t❛♠ ♥❛s ✐♠❛❣❡♥s ❡ q✉❡ sã♦ ♠♦r✜s♠♦s ❞❡ k✲á❧❣❡❜r❛s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r

  ′

  r, r ∈ R ✱ t❡♠♦s

  ′ ′

  s(r)s(r ) = (r ⊗ 1 H ⊗ 1 R )(r ⊗ 1 H ⊗ 1 R )

  

  = r(1 H ⊲ r ) ⊗ 1 H ⊗ (1 H ⊲ 1 R )1 R

  ′

  = rr ⊗ 1 H ⊗ 1 R

  ′

  = s(rr ) ❡

  ′ ′

  t(r)t(r ) = (1 R ⊗ 1 H ⊗ r)(1 R ⊗ 1 H ⊗ r )

  ′

  = 1 R (1 H ⊲ 1 R ) ⊗ 1 H ⊗ (1 H ⊲ r )r

  ′

  = 1 R ⊗ 1 H ⊗ r r

  ′ = t(r r).

  ❆❣♦r❛ t❡♠♦s

  ′ ′

  s(r)t(r ) = (r ⊗ 1 H ⊗ 1 R )(1 R ⊗ 1 H ⊗ r )

  ′

  = r(1 H ⊲ 1 R ) ⊗ 1 H ⊗ (1 H ⊲ r )1 R

  ′

  = r ⊗ 1 H ⊗ r , ♣♦r ♦✉tr♦ ❧❛❞♦✱

  ′ ′

  t(r )s(r) = (1 R ⊗ 1 H ⊗ r )(r ⊗ 1 H ⊗ 1 R )

  ′

  = 1 R (1 H ⊲ r) ⊗ 1 H ⊗ (1 H ⊲ 1 R )r

  ′ e = r ⊗ 1 H ⊗ r .

  P♦rt❛♥t♦✱ B é ✉♠ R ✲❛♥❡❧✳ ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ∆ é ♠♦r✜s♠♦ ❞❡ R

  ✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a ⊗ h ⊗ b ∈ B ❡ r ∈ R✱ t❡♠♦s ∆(r · (a ⊗ h ⊗ b)) = ∆(s(r)(a ⊗ h ⊗ b))

  = ∆((r ⊗ 1 H ⊗ 1 R )(a ⊗ h ⊗ b)) = ∆(r(1 H ⊲ a) ⊗ h ⊗ (1 H ⊲ b)1 R ) = ∆(ra ⊗ h ⊗ b) = (ra ⊗ h ⊗ 1 R ) ⊗ R (1 R ⊗ h ⊗ b)

  (1) (2)

  = (r ⊗ 1 H ⊗ 1 R )(a ⊗ h ⊗ 1 R ) ⊗ R (1 R ⊗ h ⊗ b)

  (1) (2)

  = s(r)(a ⊗ h ⊗ 1 R ) ⊗ R (1 R ⊗ h ⊗ b)

  (1) (2)

  = r · (a ⊗ h ⊗ 1 R ) ⊗ R (1 R ⊗ h ⊗ b)

  (1) (2)

  = r · ∆(a ⊗ h ⊗ b), t❛♠❜é♠ t❡♠♦s ∆((a ⊗ h ⊗ b) · r) = ∆((1 R ⊗ 1 H ⊗ r)(a ⊗ h ⊗ b))

  = ∆(1 R (1 H ⊲ a) ⊗ h ⊗ (1 H ⊲ b)r) = ∆(a ⊗ h ⊗ br)

  = (a ⊗ h ⊗ 1 R ) ⊗ R (1 R ⊗ h ⊗ br)

  (1) (2)

  = (a ⊗ h ⊗ 1 R ) ⊗ R (1 R ⊗ 1 H ⊗ r)(1 R ⊗ h ⊗ b)

  (1) (2)

  = (a ⊗ h ⊗ 1 R ) ⊗ R t(r)(1 R ⊗ h ⊗ b)

  (1) (2)

  = (a ⊗ h ⊗ 1 R ) ⊗ R (1 R ⊗ h ⊗ b) · r

  (1) (2) = ∆(a ⊗ h ⊗ b) · r.

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ε é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a ⊗ h ⊗ b ∈ B ❡ r ∈ R✱ t❡♠♦s ε((r · (a ⊗ h ⊗ b)) = ε(s(r)(a ⊗ h ⊗ b))

  = ε((r ⊗ 1 H ⊗ 1 R )(a ⊗ h ⊗ b)) = ε(r(1 H ⊲ a) ⊗ h ⊗ (1 H ⊲ b)1 R ) = ε(ra ⊗ h ⊗ b) = ra ε(h)b = rε(a ⊗ h ⊗ b), t❛♠❜é♠ t❡♠♦s

  ε((a ⊗ h ⊗ b) · r) = ε((1 R ⊗ 1 H ⊗ r)(a ⊗ h ⊗ b)) = ε(1 R (1 H ⊲ a) ⊗ h ⊗ (1 H ⊲ b)r) = ε(a ⊗ h ⊗ br) = aε(h)b r = ε(a ⊗ h ⊗ b)r.

  ▼♦str❡♠♦s q✉❡ ✈❛❧❡♠ ❛ ❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ ❝♦✉♥✐t❛❧✐❞❛❞❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ a ⊗ h ⊗ b ∈ B✱ t❡♠♦s

  (B ⊗ R ∆)∆(a ⊗ h ⊗ b) = (B ⊗ R ∆)((a⊗ h ⊗1 R )⊗ R (1 R ⊗ h ⊗b))

  

(1) (2)

  = (a⊗ h ⊗1 R ) ⊗ R [(1 R ⊗ h ⊗ 1 R ) ⊗ R (1 R ⊗ h ⊗ b)]

  (1) (2) (3)

  = [(a⊗ h ⊗1 R ) ⊗ R (1 R ⊗ h ⊗ 1 R )] ⊗ R (1 R ⊗ h ⊗ b)

  (1) (2) (3)

  = (∆⊗ R B)((a⊗ h ⊗1 R )⊗ R (1 R ⊗ h ⊗1 R ))

  

(1) (2)

= (∆ ⊗ R B)∆(a ⊗ h ⊗ b).

  ❚❛♠❜é♠ t❡♠♦s (B ⊗ R ε)∆(a ⊗ h ⊗ b) = (B ⊗ R ε)((a⊗ h ⊗1 R )⊗ R (1 R ⊗ h ⊗b))

  (1) (2)

  = (a⊗ h ⊗1 R ) · ε(1 R ⊗ h ⊗b)

  (1) (2)

  = (a⊗ h ⊗1 R ) · ε(h )b

  (1) (2)

  = t(ε(h )b)(a⊗ h ⊗1 R )

  (2) (1)

  = (1 R ⊗ 1 H ⊗ ε(h )b)(a⊗ h ⊗1 R )

  (2) (1)

  = a ⊗ h ⊗ ε(h )b

  (1) (2)

  = a ⊗ h ε(h ) ⊗ b

  (1) (2)

  = a ⊗ h ⊗ b ❡

  (ε ⊗ R B)∆(a ⊗ h ⊗ b) = (ε ⊗ R B)((a⊗ h ⊗1 R )⊗ R (1 R ⊗ h ⊗b))

  (1) (2)

  = ε(a⊗ h ⊗1 R ) · (1 R ⊗ h ⊗b)

  (1) (2)

  = aε(h ) · (1 R ⊗ h ⊗b)

  (1) (2)

  = s(aε(h ))(1 R ⊗ h ⊗b)

  (1) (2)

  = (aε(h ) ⊗ 1 H ⊗ 1 R )(1 R ⊗ h ⊗b)

  (1) (2)

  = aε(h ) ⊗ h ⊗ b

  (1) (2) = a ⊗ h ⊗ b. l

  B ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ∆(B) ⊆ B × ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r R a ⊗ h ⊗ b ∈ B

  ❡ r ∈ R✱ t❡♠♦s (a ⊗ h ⊗ 1 R )t(r) ⊗ R (1 R ⊗ h ⊗ b)

  (1) (2)

  = (a ⊗ h ⊗ 1 R )(1 R ⊗ 1 H ⊗ r) ⊗ R (1 R ⊗ h ⊗ b)

  (1) (2)

  = (a(h ⊲ 1 R ) ⊗ h ⊗ (h ⊲ r)) ⊗ R (1 R ⊗ h ⊗ b)

  (1) (2) (3) (4)

  = (aε(h ) ⊗ h ⊗ (h ⊲ r)) ⊗ R (1 R ⊗ h ⊗ b)

  

(1) (2) (3) (4)

  = (a ⊗ ε(h )h ⊗ (h ⊲ r)) ⊗ R (1 R ⊗ h ⊗ b)

  (1) (2) (3) (4)

  = (a ⊗ h ⊗ (h ⊲ r)) ⊗ R (1 R ⊗ h ⊗ b),

  (1) (2) (3)

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s (a ⊗ h ⊗ 1 R ) ⊗ R (1 R ⊗ h ⊗ b)s(r)

  (1) (2)

  = (a ⊗ h ⊗ 1 R ) ⊗ R (1 R ⊗ h ⊗ b)(r ⊗ 1 H ⊗ 1 R )

  (1) (2)

  = (a ⊗ h ⊗ 1 R ) ⊗ R (h ⊲ r ⊗ h ⊗ (h ⊲ 1 R )b)

  

(1) (2) (3) (4)

R R

  = (a ⊗ h ⊗ 1 ) ⊗ (h ⊲ r ⊗ h ⊗ ε(h )b)

  

(1) (2) (3) (4)

  = (a ⊗ h ⊗ 1 R ) ⊗ R (h ⊲ r ⊗ h ε(h ) ⊗ b)

  (1) (2) (3) (4)

  = (a ⊗ h ⊗ 1 R ) ⊗ R (h ⊲ r ⊗ h ⊗ b)

  (1) (2) (3)

  = (a ⊗ h ⊗ 1 R ) ⊗ R (h ⊲ r ⊗ 1 H ⊗ 1 R )(1 R ⊗ h ⊗ b)

  (1) (2) (3)

  = (a ⊗ h ⊗ 1 R ) ⊗ R s(h ⊲ r)(1 R ⊗ h ⊗ b)

  (1) (2) (3)

  = (a ⊗ h ⊗ 1 R ) ⊗ R (h ⊲ r) · (1 R ⊗ h ⊗ b)

  (1) (2) (3)

  = (a ⊗ h ⊗ 1 R ) · (h ⊲ r) ⊗ R (1 R ⊗ h ⊗ b)

  (1) (2) (3)

  = t((h ⊲ r))(a ⊗ h ⊗ 1 R ) ⊗ R (1 R ⊗ h ⊗ b)

  (2) (1) (3) = (a ⊗ h ⊗ (h ⊲ r)) ⊗ R (1 R ⊗ h ⊗ b).

  (1) (2) (3) l

  B ❙❡❣✉❡ q✉❡ ∆(B) ⊆ B × ✳ ❱❡r❡♠♦s ❛❣♦r❛ q✉❡ ∆ é ♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ❉❡ R

  ′

  ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r ⊗ h ⊗ r ✱ a ⊗ k ⊗ b ∈ B✱ t❡♠♦s

  ′

  ∆((r ⊗ h ⊗ r )(a ⊗ k ⊗ b))

  ′

  = ∆(r(h ⊲ a) ⊗ h k ⊗ (h ⊲ b)r )

  (1) (2) (3)

R R R

  = (r(h ⊲ a) ⊗ h k ⊗ 1 ) ⊗ (1 ⊗ h k ⊗ (h ⊲ b)r ),

  

(1) (2) (1) (3) (2) (4)

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s

  ′

  ∆(r ⊗ h ⊗ r )∆(a ⊗ k ⊗ b)

  ′

  =[(r⊗h ⊗ 1 R )⊗ R (1 R ⊗h ⊗r )][(a⊗k ⊗1 R )⊗ R (1 R ⊗k ⊗ b)]

  

(1) (2) (1) (2)

  =(r ⊗ h ⊗ 1 R )(a ⊗ k ⊗ 1 R ) ⊗ R (1 R ⊗ h ⊗ r )(1 R ⊗ k ⊗ b)

  

(1) (1) (2) (2)

  =(r(h ⊲a)⊗h k ⊗(h ⊲1 R ))⊗ R ((h ⊲1 R )⊗h k ⊗(h ⊲b)r )

  

(1) (2) (1) (3) (4) (5) (2) (6)

  =(r(h ⊲a)⊗h k ⊗(ε(h )1 R ))⊗ R (ε(h )1 R )⊗h k ⊗(h ⊲b)r )

  

(1) (2) (1) (3) (4) (5) (2) (6)

  =(r(h ⊲ a) ⊗ h k ⊗ 1 R ) ⊗ R (ε(h )1 R ) ⊗ h k ⊗ (h ⊲ b)r )

  (1) (2) (1) (3) (4) (2) (5) ′

  =(r(h ⊲ a) ⊗ h k ⊗ 1 R ) ⊗ R (1 R ⊗ h k ⊗ (h ⊲ b)r ).

  (1) (2) (1) (3) (2) (4)

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ε é ❝❛r❛❝t❡r à ❡sq✉❡r❞❛ ❡♠ (B, s)✳ ❉❡ ❢❛t♦✱ ♣❛r❛

  ′

  q✉❛✐sq✉❡r r ⊗ h ⊗ r ❡ a ⊗ k ⊗ b ∈ B✱ t❡♠♦s

  ′ ′

  ε((r ⊗ h ⊗ r )s(ε(a ⊗ k ⊗ b))) = ε((r ⊗ h ⊗ r )s(a ε(k)b))

  ′

  = ε((r ⊗ h ⊗ r )s(ab ε(k)))

  ′

  = ε((r ⊗ h ⊗ r )(ab ε(k) ⊗ 1 H ⊗ 1 R ))

  ′

  = ε(r(h ⊲ abε(k))⊗h ⊗ (h ⊲ 1 R )r )

  (1) (2) (3) ′

  = ε(r(h ⊲ ab ε(k)) ⊗ h ⊗ ε(h )r )

  (1) (2) (3) ′

  = ε(r(h ⊲ ab ε(k)) ⊗ h ⊗ r )

  (1) (2) ′

  = r(h ⊲ ab ε(k))ε(h )r

  (1) (2) ′

  = r(h ⊲ ab)ε(k)ε(h )r ,

  (1) (2) ′

  = r(h ε(h ) ⊲ ab)ε(k)r ,

  (1) (2) ′

  = r(h ⊲ ab)ε(k)r , ♣♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s

  ′ ′

  ε((r ⊗ h ⊗ r )(a ⊗ k ⊗ b)) = ε(r(h ⊲ a) ⊗ h k ⊗ (h ⊲ b)r )

  (1) (2) (3)

  ′

  = r(h ⊲ a)ε(h k)(h ⊲ b)r

  (1) (2) (3) ′

  = r(h ⊲ a)ε(h )ε(k)(h ⊲ b)r

  (1) (2) (3) ′

  = r(h ⊲ a)ε(k)(h ⊲ b)r

  (1) (2) ′

  = r(h ⊲ a)(h ⊲ b)ε(k)r

  (1) (2) ′

  = r(h ⊲ ab)ε(k)r . P❛r❛ ❝♦♥❝❧✉ír♠♦s ❛ ❞❡♠♦♥str❛çã♦✱ ♠♦str❡♠♦s q✉❡✱ ❞❡ ❢❛t♦✱ ♦ ♣r♦❞✉t♦

  ′ ′ ′

  ⊗ h ⊗ s ❛q✉✐ ❞❡✜♥✐❞♦ é ❛ss♦❝✐❛t✐✈♦✳ P❛r❛ q✉❛✐sq✉❡r r ⊗ h ⊗ s✱ r ❡ a ⊗ k ⊗ b ∈ B

  ✱ t❡♠♦s

  ′ ′ ′

  [(r ⊗ h ⊗ s)(r ⊗ h ⊗ s )](a ⊗ k ⊗ b)

  ′ ′ ′

  = (r(h ⊲ r ) ⊗ h h ⊗ (h ⊲ s )s)(a ⊗ k ⊗ b)

  (1) (2) (3) ′ ′ ′ ′ ′

  = r(h ⊲ r )(h h ⊲ a) ⊗ h h k ⊗ (h h ⊲ b)(h ⊲ s )s

  

(1) (2) (3) (4) (5)

(1) (2) (3) ′ ′ ′ ′ ′

  = r(h ⊲ (r (h ⊲ a))) ⊗ h h k ⊗ (h h ⊲ b)(h ⊲ s )s

  

(1) (2) (3) (4)

(1) (2) (3)

′ ′ ′ ′ ′

  = r(h ⊲ (r (h ⊲ a))) ⊗ h h k ⊗ (h ⊲ ((h ⊲ b)s )s,

  (1) (2) (3) (1) (2) (3)

  ♣♦r ♦✉tr♦ ❧❛❞♦✱

  ′ ′ ′

  (r ⊗ h ⊗ s)[(r ⊗ h ⊗ s )(a ⊗ k ⊗ b)]

  

′ ′ ′ ′ ′

  = (r ⊗ h ⊗ s)(r (h ⊲ a) ⊗ h k ⊗ (h ⊲ b)s )

  

(1) (2) (3)

′ ′ ′ ′ ′

  = r(h ⊲ (r (h ⊲ a))) ⊗ h h k ⊗ (h ⊲ ((h ⊲ b)s ))s.

  (1) (2) (3) (1) (2) (3) ✷✳✸✳✺ ❊①t❡♥sã♦ ❊s❝❛❧❛r

  ❙❡❥❛ B ✉♠❛ ❜✐á❧❣❡❜r❛ ❝♦♠✉t❛t✐✈❛ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k ❡ R

  ✉♠❛ á❧❣❡❜r❛ ♥❛ ❝❛t❡❣♦r✐❛ ❞♦s ♠ó❞✉❧♦s ❨❡tt❡r✲❉r✐♥❢❡❧✬❞ à ❞✐r❡✐t❛✲ ❞✐r❡✐t❛ ❞❡ B✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ R é ✉♠ B✲♠ó❞✉❧♦ á❧❣❡❜r❛ à ❞✐r❡✐t❛ ❡ ✉♠ B✲❝♦♠ó❞✉❧♦ á❧❣❡❜r❛ à ❞✐r❡✐t❛✱ t❛✐s q✉❡✱ ✈❛❧❡ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡

  

(0) (1) (0) (1)

  (a ⊳ x ) ⊗ k x (a ⊳ x ) = a ⊳ x ⊗ k a x ,

  (2) (1) (2) (1) (2) ✭✷✳✸✶✮

  ♣❛r❛ q✉❛✐sq✉❡r a ∈ R ❡ x ∈ B✳ ❊♠ q✉❡ a ⊳ x ❞❡♥♦t❛ ❛ ❛çã♦ ❞❡ B ❡♠ R

  (0) (1)

  ⊗ a ❡ a 7−→ a ❞❡♥♦t❛ ❛ ❝♦❛çã♦ ❞❡ B ❡♠ R✱ ♣❛r❛ q✉❛✐sq✉❡r a ∈ R ❡ x ∈ B✳ ❆ ❝❛t❡❣♦r✐❛ ❞♦s ♠ó❞✉❧♦s ❞❡ ❨❡tt❡r✲❉r✐♥❢❡❧✬❞ é ♣ré✲tr❛♥ç❛❞❛✳ ❆ss✉♠❛ q✉❡ R é ❝♦♠✉t❛t✐✈❛ tr❛♥ç❛❞❛✱ ♦✉ s❡❥❛✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ R✱ t❡♠♦s

  (0) (1) b (a ⊳ b ) = ab.

  ✭✷✳✸✷✮

  ❙♦❜r❡ ❡ss❛s ❝♦♥❞✐çõ❡s✱ ♦ ♣r♦❞✉t♦ s♠❛s❤ B := R#B t❡♠ ✉♠❛ ❡str✉✲ k B t✉r❛ ❞❡ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ▲❡♠❜r❡ q✉❡ R#B := R ⊗ ✱ ❝♦♠ ♠✉❧t✐♣❧✐❝❛çã♦

  (a#x)(b#y) := b(a ⊳ y ) ⊗ k xy ,

  (1) (2) ✭✷✳✸✸✮

  ♣❛r❛ q✉❛✐sq✉❡r a#x ❡ b#y ∈ B✳ ❖s ♠♦r✜s♠♦s s♦✉r❝❡ ❡ t❛r❣❡t sã♦ ❞❡✜♥✐❞♦s ♣❛r❛ t♦❞♦ a ∈ R✱ ♣♦r

  (0) (1)

  s : R −→ B, a 7−→ a #a , ♦✉ s❡❥❛✱ ♦ s♦✉r❝❡ é ❛ ❝♦❛çã♦ ❞❡ B ❡♠ R✱ ❡ t : R −→ B, a 7−→ a#1 B , r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖ ❝♦♣r♦❞✉t♦ ∆ ❡ ❛ ❝♦✉♥✐❞❛❞❡ ε✱ sã♦ ❞❡✜♥✐❞♦s✱ ♣❛r❛ t♦❞♦ a#x ∈ B✱ ♣♦r

  B B B ∆ : −→ ⊗ R a#x 7−→ (a#x ) ⊗ R (1 R #x ).

  (1) (2)

  ❡ B B B

  ε : −→ ⊗ R a#x 7−→ aε(x). ❖ ♥♦♠❡ ❡①t❡♥sã♦ ❡s❝❛❧❛r ✈❡♠ ❞♦ ❢❛t♦ q✉❡ ❛ á❧❣❡❜r❛ ❜❛s❡ k ❞❡ B é s✉❜st✐t✉í❞❛ ♣❡❧❛ á❧❣❡❜r❛ ❜❛s❡ R ❞❡ B✳ ▼♦str❡♠♦s ❡♥tã♦ q✉❡ B é ✉♠ R

  ✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ◆♦t❡ q✉❡✱ ♦ ❢❛t♦ ❞❡ R s❡r ✉♠ B✲❝♦♠ó❞✉❧♦ á❧❣❡❜r❛ à ❞✐r❡✐t❛✱ ♥♦s ❞✐③ q✉❡ ❛ ❝♦❛çã♦ ❞❡ B ❡♠ R é ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛✱ ♦✉ s❡❥❛✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ R✱ t❡♠♦s

  (0) (1) (0) (0) (1) (1)

  (ab) ⊗ k (ab) = a b ⊗ k a b ❡

  (0) (1)

  1 ⊗ k R R 1 = 1 R ⊗ k 1 R . ▼♦str❡♠♦s ♣r✐♠❡✐r♦ q✉❡ s ❡ t sã♦ ♠♦r✜s♠♦s ❞❡ á❧❣❡❜r❛ ❡ q✉❡ ❝♦♠✉t❛♠ ♥❛s ✐♠❛❣❡♥s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r ❡ a ∈ R✱ t❡♠♦s

  (0) (1)

  s(ab) = (ab) #(ab)

  (0) (0) (1) (1)

  = a b #a b , ♣♦r ♦✉tr♦ ❧❛❞♦✱

  (0) (1) (0) (1)

  s(a)s(b) = (a #a )(b #b )

  = b

  (1) (2)

  = ba

  (1)

  )#a

  (0)(1)

  (b ⊳ a

  (0)(0)

  = a

  )#a

  #a

  (1) (1)

  (b ⊳ a

  (0)

  ) = a

  (1)

  #a

  (0)

  (0)

  (1) .

  (1)

  ) = (a#x

  (2)

  )(b#1 B ) ⊗ R (1 R #x

  (1)

  ) = (a#x

  (2)

  )t(b) ⊗ R (1 R #x

  (1)

  (2)

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ∆ é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a#x ∈ B ❡ b ∈ R✱ t❡♠♦s ∆(b · (a#x)) = ∆((a#x)t(b))

  ) ⊗ R (1 R #x

  

(1)

  ), ♣♦r ♦✉tr♦ ❧❛❞♦✱ b · ∆(a#x) = b · (a#x

  (2)

  ) ⊗ R (1 R #x

  (1)

  = ∆((a#x)(b#1 B )) = ∆(b(a ⊳ 1 B )#x) = ∆(ba#x) = (ba#x

  , ♣♦r ♦✉tr♦ ❧❛❞♦✱ t(b)s(a) = (b#1 B )(a

  #a

  (0)

  = b

  (1)

  )#a

  (0)(1)

  ⊳ b

  

(0)

  (a

  (0)(0)

  (1) (2)

  (1)

  b

  (1)

  )#a

  (1) (1)

  ⊳ b

  (0)

  (a

  b

  = a

  

(0)

  

(0)

  = ba

  (1)

  ⊳ 1 B )#a

  

(0)

  )(b#1 B ) = b(a

  (1)

  #a

  ❆❣♦r❛ t❡♠♦s s(a)t(b) = (a

  (0)

  = b(a ⊳ 1 B )#1 B = ba#1 B = t(ba).

  , ✭♣♦r ✷✳✸✷✮ t❛♠❜é♠ t❡♠♦s t(a)t(b) = (a#1 H )(b#1 H )

  (1)

  b

  (1)

  #a

  (0)

  b

  )

  = (b(a ⊳ 1 B )#x ) ⊗ R (1 R #x )

  (1) (2) = (ba#x ) ⊗ R (1 R #x ). (1) (2)

  ❆❣♦r❛ ♥♦t❡ q✉❡

  (0) (1) (0) (1) (1)

  (a#x)(b #b ) = b (a ⊳ b )#xb

  (1) (2) (0)(0) (0)(1) (1)

  = b (a ⊳ b )#xb

  (0) (1)

  = ab #xb ✭♣♦r ✷✳✸✷✮. ❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s

  ∆((a#x) · b) = ∆((a#x)s(b))

  (0) (1)

  = ∆((a#x)(b #b ))

  (0) (1)

  = ∆(ab #xb )

  (0) (1) (1)

  = (ab #x b ) ⊗ R (1 R #x b )

  (1) (1) (2) (2) (0)(0) (0)(1) (1)

  = (ab #x b ) ⊗ R (1 R #x b )

  (1) (2) (0)(0) (0)(1) (1)

  = (a#x )(b #b ) ⊗ R (1 R #x b )

  (1) (2) (0) (1)

  = (a#x )s(b ) ⊗ R (1 R #x b )

  (1) (2) (0) (1)

  = (a#x ) · b ⊗ R (1 R #x b )

  (1) (2) (0) (1)

  = (a#x ) ⊗ R b · (1 R #x b )

  (1) (2) (1) (0)

  = (a#x ) ⊗ R (1 R #x b )t(b )

  

(1) (2)

(0) (1)

  = (a#x ) ⊗ R (b (1 R ⊳ 1 B )#x b )

  (1) (2) (0) (1)

  = (a#x ) ⊗ R (b #x b )

  

(1) (2)

(0) (1)

  = (a#x ) ⊗ R (1 R #x )(b #b )

  

(1) (2)

  = (a#x ) ⊗ R (1 R #x )s(b)

  

(1) (2)

  = (a#x ) ⊗ R (1 R #x )s(b)

  

(1) (2)

  = (a#x ) ⊗ R (1 R #x ) · b

  

(1) (2)

= ∆(a#x) · b.

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ε é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a#x ∈ B ❡ b ∈ R✱ t❡♠♦s ε(b · (a#x)) = ε((a#x)t(b))

  = ε((a#x)(b#1 B )) = ε(b(a ⊳ 1 B )#x)

  = ε(ba#x) = baε(x) = bε(a#x), t❛♠❜é♠ t❡♠♦s

  )) = (∆ ⊗ R

  

(1)

  )) = (a#x

  (2)

  ) ⊗ R (1 R #x

  (1)

  (B ⊗ R ε)∆(a#x) = (B ⊗ R ε)((a#x

  B )∆(a#x). ❆❣♦r❛ t❡♠♦s

  (2)

  (2)

  ) ⊗ R (1 R #x

  (1)

  B )((a#x

  ) = (∆ ⊗ R

  (2)

  ) ⊗ R (1 R #x

  

(1)

  ) · ε(1 R #x

  ) = (a#x

  (3)

  )((a#x

  (2)

  ) · (1 R #x

  

(1)

  )) = ε(a#x

  (2)

  ) ⊗ R (1 R #x

  (1)

  )∆(a#x) = (ε ⊗ R B

  

(1)

  (ε ⊗ R B

  ))(1 R #1 B ) = a#x, t❛♠❜é♠ t❡♠♦s

  (2)

  ε(x

  

(1)

  )) = (a#x

  (2)

  )s(1 R ε(x

  ) = ∆(a#x

  )) ⊗ R (1 R #x

  ε((a#x) · b) = ε((a#x)s(b)) = ε((a#x)(b

  ε(xb

  

(0)

  )) = ab

  (1)

  (ε(x)ε(b

  

(0)

  ) = ab

  (1)

  

(0)

  (1)

  ) = ab

  (1)

  #xb

  (0)

  )) = ε(ab

  (1)

  #b

  (0)

  (ε(b

  )ε(x)) = (aε(x))b = ε(a#x)b.

  (2)

  (1)

  ) ⊗ R (1 R #x

  (1)

  )) = ((a#x

  (3)

  ) ⊗ R (1 R #x

  (2)

  ) ⊗ R ((1 R #x

  ) = (a#x

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ✈❛❧❡♠ ❛ ❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ ❛ ❝♦✉♥✐t❛❧✐❞❛❞❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ a#x ∈ B✱ t❡♠♦s

  (2)

  ) ⊗ R ∆(1 R #x

  (1)

  )) = (a#x

  (2)

  ) ⊗ R (1 R #x

  (1)

  (B ⊗ R ∆)∆(a#x) = (B ⊗ R ∆)((a#x

  )

  = (1 R #x )t(ε(a#x ))

  (2) (1)

  = (1 R #x )t(aε(x ))

  (2) (1)

  = (1 R #x )(aε(x )#1 B )

  (2) (1)

  = (1 R #x ε(x ))(a#1 B )

  (2) (1)

  = (1 R #x)(a#1 B ) = a(1 R ⊳ 1 B )#x1 B = a#x. r

  B ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ∆(B) ⊆ B × R ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a#x ∈ B

  ❡ b ∈ R✱ t❡♠♦s s(b)(a#x ) ⊗ R (1 R #x )

  (1) (2) (0) (1)

  = (b #b )(a#x ) ⊗ R (1 R #x )

  (1) (2) (0) (1)

  = (a(b ⊳ x )#b x ) ⊗ R (1 R #x )

  (1) (2) (3) (0) (1)

  = (a(b ⊳ x ) #x (b ⊳ x ) ) ⊗ R (1 R #x )

  (2) (1) (2) (3) ♣♦r ✷✳✸✶

(0) (1)

  = (a#x )((b ⊳ x ) #(b ⊳ x ) ) ⊗ R (1 R #x )

  

(1) (2) (2) (3)

  = (a#x )s(b ⊳ x ) ⊗ R (1 R #x )

  (1) (2) (3)

  = (a#x ) · (b ⊳ x ) ⊗ R (1 R #x )

  (1) (2) (3)

  = (a#x ) ⊗ R (b ⊳ x ) · (1 R #x )

  (1) (2) (3)

  = (a#x ) ⊗ R (1 R #x )t(b ⊳ x )

  (1) (3) (2)

  = (a#x ) ⊗ R (1 R #x )(b ⊳ x #1 B )

  (1) (3) (2)

  = (a#x ) ⊗ R ((b ⊳ x )(1 R ⊳ 1 B )#x )

  (1) (2) (3)

  = (a#x ) ⊗ R ((b ⊳ x )#x )

  (1) (2) (3)

  = (a#x ) ⊗ R (b#1 B )(1 R #x )

  (1) (2) = (a#x ) ⊗ R t(b)(1 R #x ). (1) (2)

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ∆ é ♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a#x ❡ b#y ∈ B✱ t❡♠♦s

  ∆((a#x)(b#y)) = ∆(c(a ⊳ y )#xy )

  (1) (2)

  = (b(a ⊳ y )#x y ) ⊗ R (1 R #x y ),

  (1) (1) (2) (2) (3)

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ ∆(a#x)∆(b#y) = [(a#x ) ⊗ R (1 R #x )][(b#y ) ⊗ R (1 R #y )]

  (1) (2) (1) (2)

  = (a#x )(b#y ) ⊗ R (1 R #x )(1 R #y )

  (1) (1) (2) (2)

  = (b(a ⊳ y )#x y ) ⊗ R (1 R ⊳ y #x y )

  (1) (1) (2) (3) (2) (4)

  = (b(a ⊳ y )#x y ) ⊗ R (1 R ε(y )#x y )

  (1) (1) (2) (3) (2) (4)

  = (b(a ⊳ y )#x y ) ⊗ R (1 R #x ε(y )y )

  (1) (1) (2) (2) (3) (4) = (b(a ⊳ y )#x y ) ⊗ R (1 R #x y ). (1) (1) (2) (2) (3)

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ε é ❝❛r❛❝t❡r à ❞✐r❡✐t❛ ❡♠ (B, s)✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a#x ❡ b#y ∈ B✱ t❡♠♦s ε(s(ε(a#x))b#y) = ε(s(aε(x))(b#y))

  = ε(s(a)(b#y)ε(x))

  (0) (1)

  = ε((a #a )(b#y))ε(x)

  (0) (1)

  = ε(b(a ⊳ y )#a y )ε(x)

  (1) (2) (0) (1)

  = b(a ⊳ y )ε(a y )ε(x)

  (1) (2) (0) (1)

  = b(a ⊳ y )ε(a )ε(y )ε(x)

  (1) (2)

  = b(a ⊳ y)ε(x), ♣♦r ♦✉tr♦ ❧❛❞♦✱

  ε((a#x)(b#y)) = ε(b(a ⊳ y )#x y )

  (1) (2)

  = b(a ⊳ y )ε(x y )

  (1) (2)

  = b(a ⊳ y )ε(x)ε(y )

  (1) (2) = b(a ⊳ y)ε(x).

  P♦rt❛♥t♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ B é ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳

  ✷✳✹ ❆ ❈❛t❡❣♦r✐❛ ▼♦♥♦✐❞❛❧ ❞❡ ▼ó❞✉❧♦s

  ❙❛❜❡♠♦s q✉❡ ✉♠❛ á❧❣❡❜r❛ B s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k✱ ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❜✐á❧❣❡❜r❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ❝❛t❡❣♦r✐❛ ❞❡ B✲♠ó❞✉❧♦s à ❞✐r❡✐t❛ ✭♦✉ à ❡sq✉❡r❞❛ ✮ é ♠♦♥♦✐❞❛❧✱ t❛❧ q✉❡ ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ M B −→ M k

  é ❡str✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧✱ ✈❡r ❬✷✸❪✳ ◆♦ss♦ ♣ró①✐♠♦ t❡♦r❡♠❛ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡st❡ ❢❛t♦ ♣❛r❛ ❜✐❛❧❣❡❜ró✐❞❡s✳ ❚❛❧ ❣❡♥❡r❛❧✐③❛çã♦✱ ❢♦✐ ❢❡✐t❛ ♣♦r ❙❝❤❛✉❡♥❜✉r❣ ❡♠ ❬✷✻❪✳ ❚❡♦r❡♠❛ ✷✳✷✶ ❙❡❥❛♠ R ✉♠❛ á❧❣❡❜r❛ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k ❡ e (B, s, t)

  ✉♠ R ✲❛♥❡❧✳ ❊♥tã♦✱ ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿ (1) B

  é ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛❀ (2) B

  ❆ ❝❛t❡❣♦r✐❛ M ❞♦s B✲♠ó❞✉❧♦s à ❞✐r❡✐t❛ é ♠♦♥♦✐❞❛❧✱ t❛❧ q✉❡ ♦ B −→ R R M ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ U : M é ❡str✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧✳

  ❉❡♠♦♥str❛çã♦✿ (1) ⇒ (2) Pr✐♠❡✐r❛♠❡♥t❡✱ ♥♦t❡ q✉❡✱ s❡ M é ✉♠ B✲ ♠ó❞✉❧♦ à ❞✐r❡✐t❛✱ ❡♥tã♦ M é ✉♠ R✲❜✐♠ó❞✉❧♦ ❝♦♠ ❡str✉t✉r❛ ❞❛❞❛ ♣♦r

  ′ ′ ′

  r · m · r := (m ⊳ s(r )) ⊳ t(r) ∈ R B R N ∈ M B R N ✱ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M ❡ r, r ✳ ❆❣♦r❛✱ ♣❛r❛ q✉❛✐sq✉❡r M, N ∈ M ✱ t❡♠♦s M ⊗ ✳ ❉❡ ❢❛t♦✱ M ⊗ ♣♦ss✉✐ ❡str✉t✉r❛ ❞❡ B✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✱ ❞❡✜♥✐❞❛ ♣♦r (m ⊗ R n) ⊳ b := m ⊳ b ⊗ R n ⊳ b . R N (1) (2)

  ❆ss✐♠✱ ♣❛r❛ q✉❛✐sq✉❡r m ⊗ n ∈ M ⊗ ❡ a, b ∈ B✱ t❡♠♦s ((m ⊗ n) ⊳ a) ⊳ b = (m ⊳ a ⊗ n ⊳ a ) ⊳ b

  (1) (2)

  = (m ⊳ a ) ⊳ b ⊗ (n ⊳ a ) ⊳ b

  (1) (1) (2) (2)

  = m ⊳ a b ⊗ n ⊳ a b

  (1) (1) (2) (2)

  = m ⊳ (ab) ⊗ n ⊳ (ab)

  (1) (2) = (m ⊗ n) ⊳ ab.

  ❚❛♠❜é♠ t❡♠♦s ♣❡❧♦ ▲❡♠❛ ✶✳✹✻ q✉❡ R é ✉♠ B✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✱ ♣♦✐s ε

  é ❝❛r❛❝t❡r à ❞✐r❡✐t❛ ❡♠ (B, s)✳ ❆ ❛çã♦ ❞❡ B ❡♠ R é ❞❛❞❛ ♣♦r r ⊳ b = ε(s(r)b) ✱ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R ❡ b ∈ B✳ ❆❣♦r❛✱ ❞❡✜♥✐♠♦s ♦s

  ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s ❝❛♥ô♥✐❝♦s a M,N,P : (M ⊗ R N )⊗ R P −→ M ⊗ R (N ⊗ R P ), (m⊗n)⊗p 7−→ m⊗(n⊗p), l M : R ⊗ R M −→ M, (r ⊗ m) 7−→ r · m

  ❡ r M : M ⊗ R R −→ M, (m ⊗ r) 7−→ m · r. ❱❡r❡♠♦s q✉❡ sã♦ ♠♦r✜s♠♦s ❞❡ B✲♠ó❞✉❧♦s à ❞✐r❡✐t❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r (m ⊗ n) ⊗ p ∈ (M ⊗ N) ⊗ P ❡ b ∈ B✱ t❡♠♦s a M,N,P (((m ⊗ n) ⊗ p) ⊳ b) = a M,N,P ((m ⊗ n) ⊳ b ⊗ p ⊳ b )

  (1) (2)

  = a M,N,P ((m ⊳ b ⊗ n ⊳ b ) ⊗ p ⊳ b )

  (1) (2) (3)

  = m ⊳ b ⊗ (n ⊳ b ⊗ p ⊳ b )

  (1) (2) (3)

  = m ⊳ b ⊗ ((n ⊗ p) ⊳ b )

  (1) (2)

  = (m ⊗ (n ⊗ p)) ⊳ b = a M,N,P ((m ⊗ n) ⊗ p) ⊳ b. ❆❣♦r❛ ♣❛r❛ q✉❛✐sq✉❡r r ⊗ m ∈ R ⊗ M ❡ b ∈ B✱ t❡♠♦s l M ((r ⊗ m) ⊳ b) = l M (r ⊳ b ⊗ m ⊳ b )

  (1) (2)

  = l M (ε(s(r)b ) ⊗ m ⊳ b )

  (1) (2)

  = ε(s(r)b

  s(ε(b

  (2)

  )) = (m ⊳ s(r)b

  (1)

  ) · ε(b

  (2)

  )) = (m ⊳ s(r)b

  (1)

  ) ⊳ s(ε(b

  (2)

  )) = m ⊳ s(r)b

  (1)

  (2)

  (1)

  )) = m ⊳ s(r)b = (m ⊳ s(r)) ⊳ b = (m · r) ⊳ b = r M (m ⊗ r) ⊳ b.

  ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ sã♦ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s✱ ♣♦✐s s❡❣✉❡ s✐♠♣❧❡s♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❝❛❞❛ ✐s♦♠♦r✜s♠♦ ❡ ❞♦ ❢❛t♦ q✉❡ sã♦ ❛♣❧✐❝❛❞♦s ❡♠ ♠♦r✲ ✜s♠♦s ❞❡ B✲♠ó❞✉❧♦s✳ ◗✉❡ ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ U : M B −→ R

  M R é ❡str✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧ é ❝❧❛r♦✱ ♣♦✐s ❜❛st❛ ❞❡✜♥✐r♠♦s ϕ

  : R −→ R ❡ φ M,N : M ⊗ R N −→ M ⊗ R N ❝♦♠♦ ✐❞❡♥t✐❞❛❞❡s q✉❡ ♦s ❞✐❛❣r❛♠❛s ♥❡❝❡ssár✐♦s✱ ❝♦♠✉t❛♠ tr✐✈✐❛❧♠❡♥t❡✳ (1) ⇐ (2)

  ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ ♦ R e ✲❛♥❡❧ (B, s, t) ♣♦ss✉✐ ❡str✉t✉r❛ ❞❡ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❉❡ ❢❛t♦✱ ♣r✐♠❡✐r♦ ✈❛♠♦s ♠✉♥✐r B ❝♦♠ ❡str✉t✉r❛ ❞❡ R✲❝♦❛♥❡❧✳ ◆♦t❡ q✉❡ B é B✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❝♦♠ ❛çã♦ ❞❛❞❛ ♣❡❧♦ ♣r♦❞✉t♦✱ ❡♥tã♦ t❡♠♦s q✉❡ B ⊗ R

  B é B✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✱

  ♣♦✐s M B é ♠♦♥♦✐❞❛❧✱ t❛❧ q✉❡ ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ M B −→ R M R

  é ❡str✐t♦✱ ❡ ♣❡❧♦ ♠❡s♠♦ ♠♦t✐✈♦ R é B✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❆❣♦r❛✱ ❞❡✜♥❛ ∆ : B −→ B ⊗ R B

  ✱ t❛❧ q✉❡ ∆(b) := (1 B ⊗ 1 B ) ⊳ b = b

  (1)

  ⊗ b

  (2) ✳

  ) · ε(t(r)b

  )) = (m ⊳ b

  (1)

  t(ε(b

  ) · (m ⊳ b

  (2)

  ) = ε(b

  

(1)

  ) · (m ⊳ t(r)b

  (2)

  ) = (m ⊳ t(r)b

  (2)

  ) ⊳ t(ε(b

  (1)

  )) = m ⊳ t(r)b

  (2)

  (1)

  (2)

  )) = m ⊳ t(r)b = (m ⊳ t(r)) ⊳ b = (r · m) ⊳ b = l M (r ⊗ m) ⊳ b

  ❡ t❛♠❜é♠ r M ((m ⊗ r) ⊳ b) = r M (m ⊳ b

  (1)

  ⊗ r ⊳ b

  (2)

  ) = r M (m ⊳ b

  (1)

  ⊗ ε(s(r)b

  (2)

  )) = (m ⊳ b

  (1)

  ) · ε(s(r)b

  ❆✜r♠❛çã♦✿ ❙❡❥❛♠ M ❡ N B✲♠ó❞✉❧♦s à ❞✐r❡✐t❛✳ ❊♥tã♦✱ ♣❛r❛ q✉❛✐sq✉❡r m ⊗ n ∈ M ⊗ R N ❡ b ∈ B✱ t❡♠♦s

  (m ⊗ n) ⊳ b = m ⊳ b

  ) = (1 B ⊳ b

  ) = a B,B,B ((1 B ⊗ 1 B ) ⊳ b

  (1)

  ⊗ b

  (2)

  ) = a B,B,B (((1 B ⊗ 1 B ) ⊗ 1 B ) ⊳ b) = a B,B,B ((1 B ⊗ 1 B ) ⊗ 1 B ) ⊳ b = (1 B ⊗ (1 B ⊗ 1 B )) ⊳ b = (1 B ⊳ b

  (1)

  ⊗ (1 B ⊗ 1 B ) ⊳ b

  (2)

  (1)

  ) ⊗ b

  ⊗ (b

  (2)(1)

  ⊗ b

  (2)(2)

  )) = (b

  

(1)

  ⊗ ∆(b

  (2)

  )) = (B ⊗ R ∆)∆(b). ❆❣♦r❛✱ ♣❡❧♦ ▲❡♠❛ ✶✳✹✻✱ ❝♦♠♦ R é B✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✱ t❡♠♦s q✉❡ ❡①✐st❡ ε : B −→ R

  (2)

  (1)(2)

  (1)

  = (ρ M m ⊗ R ρ N n )(b

  ⊗ R n ⊳ b

  (2) .

  ✭✷✳✸✹✮ ❉❡ ❢❛t♦✱ ♣r✐♠❡✐r♦ ♣❛r❛ m ∈ M✱ ❞❡✜♥❛ ρ M m : B −→ M, b 7−→ m⊳b ✱ ♣❛r❛ t♦❞♦ b ∈ B✳ ▼♦str❡♠♦s q✉❡ ρ M m é ♠♦r✜s♠♦ ❞❡ B✲♠ó❞✉❧♦s à ❞✐r❡✐t❛✳

  ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ B✱ t❡♠♦s ρ M m (ab) = m ⊳ ab

  = (m ⊳ a) ⊳ b = ρ M m (a) ⊳ b. P♦rt❛♥t♦✱ ♣❛r❛ m ∈ M ❡ n ∈ N✱ t❡♠♦s q✉❡

  ρ M m ⊗ R ρ N n : B ⊗ R B −→ M ⊗ R N, a ⊗ b 7−→ m ⊳ a ⊗ n ⊳ b, é ♠♦r✜s♠♦ ❞❡ B✲♠ó❞✉❧♦s à ❞✐r❡✐t❛✳ ❙❡❣✉❡ ❡♥tã♦ m ⊳ b

  (1)

  ⊗ n ⊳ b

  (2)

  (1)

  ⊗ b

  ⊗ b

  (2)

  ) = (ρ M m ⊗ R ρ N n )((1 B ⊗ 1 B ) ⊳ b) = ((ρ M m ⊗ R ρ N n )(1 B ⊗ 1 B )) ⊳ b = (m ⊳ 1 B ⊗ n ⊳ 1 B ) ⊳ b = (m ⊗ n) ⊳ b.

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ a B,B,B ((∆ ⊗ B)∆(b)) = (B ⊗ ∆)∆(b) ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ b ∈ B✱ t❡♠♦s a B,B,B ((∆ ⊗ B)∆(b)) = a B,B,B (∆(b

  (1)

  ) ⊗ b

  (2)

  ) = a B,B,B ((b

  (1)(1)

  ✱ ❝❛r❛❝t❡r à ❞✐r❡✐t❛ ❡♠ (B, s)✱ t❛❧ q✉❡ ε(b) = 1 R ⊳ b ✳ ▼♦s✲ tr❡♠♦s ❡♥tã♦✱ q✉❡ ✈❛❧❡ ❛ ❝♦✉♥✐t❛❧✐❞❛❞❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ b ∈ B✱ t❡♠♦s r B ((B ⊗ ε)∆(b)) = r B ((B ⊗ ε)(b

  (1)

  ) ⊗ b

  ∆(b · r) = ∆(bs(r)) = (1 B ⊗ 1 B ) ⊳ bs(r) = ((1 B ⊗ 1 B ) ⊳ b) ⊳ s(r) = ∆(b) ⊳ s(r) = ∆(b) · r.

  ∆(r · b) = ∆(bt(r)) = (1 B ⊗ 1 B ) ⊳ bt(r) = ((1 B ⊗ 1 B ) ⊳ b) ⊳ t(r) = ∆(b) ⊳ t(r) = r · ∆(b), t❛♠❜é♠ t❡♠♦s

  ❱❛♠♦s ✈❡r ❛❣♦r❛ q✉❡ ∆ ❡ ε sã♦ ♠♦r✜s♠♦s ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r b ∈ B ❡ r ∈ R✱ t❡♠♦s

  ) = l B ((1 B ⊗ 1 R ) ⊳ b) = l B (1 B ⊗ 1 R ) ⊳ b = 1 B ⊳ b = b.

  (2)

  ⊗ b

  (1)

  ) = l B (1 R ⊳ b

  (2)

  (1)

  ⊗ b

  ) = r B ((1 B ⊗ 1 R ) ⊳ b) = r B (1 B ⊗ 1 R ) ⊳ b = 1 B ⊳ b = b, t❛♠❜é♠ t❡♠♦s l B ((ε ⊗ B)∆(b)) = l B (ε(b

  (2)

  ⊗ 1 R ⊳ b

  (1)

  )) = r B (b

  (2)

  ⊗ ε(b

  (1)

  )) = r B (b

  (2)

  ❆❣♦r❛ ε(r · b) = ε(bt(r))

  = 1 R ⊳ bt(r) = (1 R ⊳ b) ⊳ t(r) = ε(b) ⊳ t(r) = rε(b), t❛♠❜é♠

  ε(b · r) = ε(bs(r)) = 1 R ⊳ bs(r) = (1 R ⊳ b) ⊳ s(r) = ε(b) ⊳ s(r) = ε(b)r. r

  B ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ∆(B) ⊆ B × R ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r b ∈ B ❡ r ∈ R✱ t❡♠♦s s(r)b ⊗ b = s(r) ⊳ b ⊗ 1 B ⊳ b

  (1) (2) (1) (2)

  = (s(r) ⊗ 1 B ) ⊳ b = (1 B · r ⊗ 1 B ) ⊳ b = (1 B ⊗ r · 1 B ) ⊳ b = (1 B ⊗ t(r)) ⊳ b = 1 B ⊳ b ⊗ t(r) ⊳ b

  (1) (2) = b ⊗ t(r)b . (1) (2)

  ❋❛❧t❛ ♠♦str❛r♠♦s ❛❣♦r❛ q✉❡ ∆ é ♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r

  a, b ∈ B ✱ t❡♠♦s

  ∆(ab) = (1 B ⊗ 1 B ) ⊳ ab = ((1 B ⊗ 1 B ) ⊳ a) ⊳ b = (a ⊗ a ) ⊳ b

  (1) (2)

  = a ⊳ b ⊗ a ⊳ b

  (1) (1) (2) (2)

  = a b ⊗ a b

  

(1) (1) (2) (2)

  = (a ⊗ a )(b ⊗ b )

  (1) (2) (1) (2) e = ∆(a)∆(b).

  P♦rt❛♥t♦✱ t❡♠♦s q✉❡ ♦ R ✲❛♥❡❧ (B, s, t) ♣♦ss✉✐ ❡str✉t✉r❛ ❞❡ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❈♦♥❝❧✉í♠♦s ❛ss✐♠✱ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛✳

  ✷✳✺ ❆ ❈❛t❡❣♦r✐❛ ▼♦♥♦✐❞❛❧ ❞❡ ❈♦♠ó❞✉❧♦s

  ❙❡❥❛ B ✉♠❛ ❜✐á❧❣❡❜r❛ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k✳ ❆ ❝❛t❡❣♦r✐❛ ❞♦s ❝♦♠ó❞✉❧♦s à ❞✐r❡✐t❛ s♦❜r❡ B✱ ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ♠♦♥♦✐❞❛❧✱ t❛❧ k q✉❡ ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ ♣❛r❛ M é ❡str✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧✳ P❛r❛ R✲ ❜✐❛❧❣❡❜ró✐❞❡s à ❞✐r❡✐t❛ B✱ ❡ss❡ r❡s✉❧t❛❞♦ ❝♦♥t✐♥✉❛ s❡♥❞♦ ✈á❧✐❞♦✳ P♦ré♠✱ ❛♦ ❝♦♥trár✐♦ ❞♦ ❝❛s♦ ♣❛r❛ ♠ó❞✉❧♦s✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❢✉♥t♦r ❡sq✉❡❝✐✲ ♠❡♥t♦ ♥ã♦ é ❡✈✐❞❡♥t❡✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ♥ã♦ é ❝❧❛r♦ q✉❡ ♠♦r✜s♠♦s ❞❡ B✲♠ó❞✉❧♦s s❡❥❛♠ ♠♦r✜s♠♦s ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❖ ♣ró①✐♠♦ ❧❡♠❛ ♥♦s ❣❛r❛♥t❡ ✐st♦✳ ❚❛❧ ❧❡♠❛ ❢♦✐ ♣r♦✈❛❞♦ ❡♠ ❬✶✷❪ ♥♦ ❝♦♥t❡①t♦ ❞❡ ❝♦❛❧❣❡❜ró✐❞❡s✳ ❆♥t❡s✱ ♣r❡❝✐s❛♠♦s ❞❡ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s✳ ❉❡✜♥✐çã♦ ✷✳✷✷ ❙❡❥❛♠ R ❡ S ❞✉❛s á❧❣❡❜r❛s s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k k R

  ✳ ❯♠ S|R✲❝♦❛♥❡❧ é ✉♠ S ⊗ ✲❜✐♠ó❞✉❧♦ C✱ ❥✉♥t♦ ❝♦♠ ✉♠❛ ❡str✉t✉r❛

  ′

  ∈ S ❞❡ R✲❝♦❛♥❡❧ (C, ∆, ε)✱ t❛✐s q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r s, s ❡ c ∈ C✱ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿ R ) · c · (s ⊗ 1 R )) = c · (s ⊗ 1 R ) ⊗ R (s ⊗ 1 R ) · c ′ ′

  ✭✐✮ ∆((s ⊗ 1 (1) (2) ❀ R ) · c ⊗ R c = c ⊗ R c · (s ⊗ 1 R ) ✭✐✐✮ (s ⊗ 1 (1) (2) (1) (2) ✳

  ▼♦r✜s♠♦s ❞❡ S|R✲❝♦❛♥é✐s✱ sã♦ ♠♦r✜s♠♦ ❞❡ R✲❝♦❛♥é✐s ❡ ❛❞✐❝✐♦♥❛❧✲ ♠❡♥t❡✱ ♠♦r✜s♠♦s ❞❡ S✲❜✐♠ó❞✉❧♦s✳

  ❚♦❞♦ S|R✲❝♦❛♥❡❧ C ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ❞❡ S✲❜✐♠ó❞✉❧♦ ❡ ✉♠❛ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦✱ q✉❡ ✐r❡♠♦s ❞❡♥♦t❛r✱ ♣❛r❛ q✉❛✐sq✉❡r c ∈ C✱ s ∈ S

  ❡ r ∈ R✱ ♣♦r τ S (s)c := (s ⊗ 1 R ) · c, c σ S (s) := c · (s ⊗ 1 R )

  ✭✷✳✸✺✮ ❡ σ R (r)c := (1 S ⊗ r) · c, c τ R (r) := c · (1 S ⊗ r).

  ✭✷✳✸✻✮ ❆❣♦r❛ t❡♠♦s q✉❡ C é ✉♠ R✲❝♦❛♥❡❧ ❝♦♠ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦ ❡♠

  ′

  C ⊗ R C ∈ R ❞❛❞❛✱ ♣❛r❛ q✉❛✐sq✉❡r r, r ❡ c, ∈ C✱ ♣♦r

  ′ ′ ′ ′ r(c ⊗ R c )r := σ R (r)c ⊗ R c τ R (r ). R C

  ❚❛♠❜é♠ t❡♠♦s q✉❡ C ⊗ ♣♦ss✉✐ ❡str✉t✉r❛ ❞❡ S✲❜✐♠ó❞✉❧♦✱ ❞❛❞❛ ♣❛r❛

  ′ ′

  ∈ S ∈ C q✉❛✐sq✉❡r s, s ❡ c, c ✱ ♣♦r

  ′ ′ ′ ′ s(c ⊗ R c )s := c σ S (s ) ⊗ R τ S (s)c .

  ❘❡❡s❝r❡✈❡♥❞♦ ❛ ❝♦♥❞✐çã♦ (i) ❞❡ S|R✲❝♦❛♥❡❧✱ t❡♠♦s

  

′ ′

  ∆(τ S (s)c σ S (s )) = c σ S (s ) ⊗ R τ S (s)c ,

  (1) (2)

  ♦✉ s❡❥❛✱ ∆ é ♠♦r✜s♠♦ ❞❡ S✲❜✐♠ó❞✉❧♦✳ ❚❛♠❜é♠ r❡❡s❝r❡✈❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦✉♥✐t❛❧✐❞❛❞❡✱ t❡♠♦s c = σ R (ε(c ))c = c τ R (ε(c )).

  (1) (2) (1) (2)

  ❆ ❝♦♥❞✐çã♦ (ii) ♥♦s ❣❛r❛♥t❡✱ ♣❛r❛ q✉❛✐sq✉❡r s ∈ S ❡ c ∈ C✱ q✉❡ ε(τ S (s)c) = ε(c σ S (s)).

  ✭✷✳✸✼✮ ❉❡ ❢❛t♦✱ t❡♠♦s

  ε(τ S (s)c) = ε(τ S (s)c τ R (ε(c )))

  (1) (2)

  = ε(τ S (s)c )ε(c )

  (1) (2) S

  = ε(c )ε(c σ (s))

  (1) (2) ♣♦r (ii)

  = ε(σ R (ε(c ))c σ S (s))

  (1) (2) = ε(cσ S (s)).

  ❯s❛♥❞♦ ❛ ❝♦✉♥✐t❛❧✐❞❛❞❡ ❡ ♦ ❢❛t♦ q✉❡ ∆ é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s ❡ ❞❡ S✲❜✐♠ó❞✉❧♦s✱ s❡❣✉❡ ♥❛ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦✱ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ s❡rã♦ út❡✐s ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ♣ró①✐♠♦ ❧❡♠❛✳ Pr♦♣♦s✐çã♦ ✷✳✷✸ ❙❡❥❛ C ✉♠ S|R✲❝♦❛♥❡❧✳ ❊♥tã♦✱ t❡♠♦s

  (1) c σ S (s) = σ R (ε(c σ S (s)))c S (s)c = c τ R (ε(τ S (s)c ));

  (1) (2) ❡ τ (1) (2) ′ ′ ′

  (2) ε(τ S (s)c σ S (s )) = ε(c σ S (s ))ε(τ S (s)c ) ∈

  (1) (2) ✱ ♣❛r❛ q✉❛✐sq✉❡r s, s

  S ❡ c ∈ C✳

  ❉❡♠♦♥str❛çã♦✿ (1) ◆♦t❡ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r s ∈ S ❡ c ∈ C✱ t❡♠♦s ∆(c σ S (s)) = c σ S (s) ⊗ R c ,

  (1) (2)

  ❡ ∆(τ S (s)c) = c ⊗ R τ S (s)c .

  (1) (2)

  ❙❡❣✉❡ ♣♦rt❛♥t♦✱ q✉❡ c σ S (s) = σ R (ε((c σ S (s)) ))(cσ S (s)) = σ R (ε(c σ S (s)))c

  (1) (2) (1) (2) ✭✷✳✸✽✮

  ❡ q✉❡ τ S (s)c = (τ S (s)c) τ R (ε((τ S (s)c) )) = c τ R (ε(τ S (s)c ))

  (1) (2) (1) (2) ✭✷✳✸✾✮ ′

  (2) ∈ S P❛r❛ q✉❛✐sq✉❡r s, s ❡ c ∈ C✱ t❡♠♦s S S S S ′ ′

  ε(τ (s)c σ (s )) = ε(τ (s )τ (s)c) ✭♣♦r ✷✳✸✼✮

  ′

  = ε(τ S (s s)c)

  

  = ε(c τ R (ε(τ S (s s)c )))

  (1) (2) ✭♣♦r ✷✳✸✾✮ ′

  = ε(c )ε(τ S (s )τ S (s)c )

  (1) (2) ′

  = ε(c )ε(τ S (s)c σ S (s ))

  (1) (2) ′

  = ε(c )ε(σ R (ε(c σ S (s )))τ S (s)c )

  (1) (2) (3) ✭♣♦r ✷✳✸✽✮ ′

  = ε(c )ε(c σ S (s ))ε(τ S (s)c )

  (1) (2) (3) ′

  = ε(σ R (ε(c ))c σ S (s ))ε(τ S (s)c )

  (1) (2) (3) ′

  = ε(c σ S (s ))ε(τ S (s)c ).

  

(1) (2)

  ✭✷✳✹✵✮ Pr♦♣♦s✐çã♦ ✷✳✷✹ ❙❡❥❛ B ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❊♥tã♦ B é ✉♠ R|R

  ✲❝♦❛♥❡❧✳ ❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ ❞❡✜♥✐♠♦s ✉♠❛ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦✱

  ′

  ∈ R ♣❛r❛ q✉❛✐sq✉❡r r, r ❡ b ∈ B✱ ♣♦r

  ′ ′

  σ R (r)bτ R (r ) := bs(r )t(r), ♦✉ s❡❥❛✱ é ❛ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦ ♥♦ R✲❝♦❛♥❡❧ (B, ∆, ε)✳ ❉❡✜♥✐♠♦s t❛♠❜é♠✱ ♦✉tr❛ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦ ♣♦r S S ′ ′ τ (r)bσ (r ) := s(r)t(r )b.

  P❛r❛ ♠♦str❛r♠♦s q✉❡ B é ✉♠ R|R✲❝♦❛♥❡❧✱ ❜❛st❛ ♠♦str❛r♠♦s ❛s ❝♦♥❞✐✲ çõ❡s (i) ❡ (ii) ❞❛ ❞❡✜♥✐çã♦✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ r ∈ R✱ ♥♦t❡ q✉❡

  ∆(t(r)) = ∆(r · 1 B ) = r · (1 B ⊗ 1 B ) = r · 1 B ⊗ 1 B = t(r) ⊗ R

  1 R , ❛♥❛❧♦❣❛♠❡♥t❡✱ ∆(s(r)) = 1 R ⊗ R s(r).

  ′

  ∈ R ❉❡ss❛ ❢♦r♠❛✱ ♣❛r❛ q✉❛✐sq✉❡r r, r ❡ b ∈ B✱ t❡♠♦s

  

′ ′

  ∆(τ S (r)b σ(r )) = ∆(s(r)t(r )b)

  

  = t(r )b ⊗ R s(r)b

  (1) (2) ′

  = b σ S (r ) ⊗ R τ S (r)b .

  (1) (2)

  t❛♠❜é♠ t❡♠♦s τ S (r)b ⊗ R b = s(r)b ⊗ R b

  (1) (2) (1) (2)

  = b ⊗ R t(r)b

  (1) (2) = b ⊗ R b σ S (r). (1) (2)

  ❆❣♦r❛ ❡st❛♠♦s ♣r♦♥t♦s ♣❛r❛ ♣r♦✈❛r ♦ ♣ró①✐♠♦ ❧❡♠❛✳ ▲❡♠❛ ✷✳✷✺ ❙❡❥❛♠ R✱ S á❧❣❡❜r❛s s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k ❡ C ✉♠ M S|R )

  ✲❝♦❛♥❡❧✳ ❊♥tã♦✱ t♦❞♦ ❝♦♠ó❞✉❧♦ (M, ρ s♦❜r❡ ♦ R✲❝♦❛♥❡❧ C✱ ♣♦❞❡ s❡r ❡q✉✐♣❛❞♦ ❝♦♠ ✉♠❛ ú♥✐❝❛ ❡str✉t✉r❛ ❞❡ S✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛✱ t❛❧ q✉❡ M ρ (m) R C

  ♣❡rt❡♥❝❡ ❛♦ ❝❡♥tr♦ ❞♦ S✲❜✐♠ó❞✉❧♦ M ⊗ ✱ ♣❛r❛ t♦❞♦ m ∈ M

  ✳ ❚❛♠❜é♠✱ t♦❞♦ ♠♦r✜s♠♦ ❞❡ C✲❝♦♠ó❞✉❧♦s é ✉♠ ♠♦r✜s♠♦ ❞❡ S✲R✲ ❜✐♠ó❞✉❧♦s✳ ❉❡♠♦♥str❛çã♦✿ ❉❡✜♥✐♠♦s ❛ ❡str✉t✉r❛ ❞❡ S✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ ❡♠ M

  ✱ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M ❡ a ∈ S✱ ♣♦r

  (0) (1) R S s · m := m τ (ε(τ (s)m )).

  ✭✷✳✹✶✮ ▼♦str❡♠♦s q✉❡ ❞❡ ❢❛t♦ é ❛çã♦ à ❡sq✉❡r❞❛✳ P❛r❛ q✉❛✐sq✉❡r a, b ∈ S ❡ m ∈ M

  ✱ t❡♠♦s

  

(0) (1)

  a · (b · m) = a · (m τ R (ε(τ S (b)m )))

  (0)(0) (0)(1) (1)

  = m τ R (ε(τ S (a)m τ R (ε(τ S (b)m ))))

  (0) (1) (1)

  = m τ R (ε(τ S (a)m τ R (ε(τ S (b)m ))))

  (1) (2) (0) (1) (1)

  = m τ R (ε(τ S (a)m )ε(τ S (b)m ))

  (1) (2) (0) (1) (1)

  = m τ R (ε(m σ S (a))ε(τ S (b)m ))

  (1) (2) (0) (1)

  = m τ R (ε(τ S (b)m σ S (a))) ✭♣♦r ✷✳✹✵✮

  (0) (1)

  = m τ R (ε(τ S (a)τ S (b)m ))

  (0) (1)

  = m τ R (ε(τ S (ab)m )) = ab · m. R C ❆❣♦r❛ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ❡str✉t✉r❛ ❞❡ S✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ ❡♠ M ⊗ ✱ R M

  c) := m ⊗ R τ S (a)c : M −→ M ⊗ R C ❞❛❞❛ ♣♦r a(m ⊗ ✱ t❡♠♦s q✉❡ ρ

  é S✲❧✐♥❡❛r ❝♦♠ r❡s♣❡✐t♦ ❛ ❛çã♦ ✷✳✹✶✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M ❡ a ∈ S ✱ t❡♠♦s

  (0) (1) (0) (1) (1)

  m ⊗ R τ S (a)m = m ⊗ R m τ R (ε(τ S (a)m ))

  (1) (2) ✭♣♦r ✷✳✸✽✮ (0)(0) (0)(1) (1)

  = m ⊗ R m τ R (ε(τ (a)m ))

  (0)(0) (0)(1) (1)

  = (m ⊗ R m )τ R (ε(τ S (a)m )) M

  (0) (1)

  = ρ (m )τ R (ε(τ (a)m )) M

  (0) (1)

  = ρ (m τ R (ε(τ S (a)m ))) M = ρ (a · m). R C ❆❣♦r❛ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ❡str✉t✉r❛ ❞❡ S✲❜✐♠ó❞✉❧♦ ❡♠ M ⊗ ✱ ❞❛❞❛ ♣♦r a(m⊗ R c)b := a·m⊗ R c σ S (b)

  ✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ S✱ m ∈ M ❡ c ∈ C✱ t❡♠♦s q✉❡

  (0) (1) (0) (1)

  a(m ⊗ R m ) = a · m ⊗ R m

  (0)(0) (0)(1) (1)

  = m τ R (ε(τ S (a)m )) ⊗ R m

  (0) (1) (1)

  = m τ R (ε(τ S (a)m )) ⊗ R m

  (1) (2) (0) (1) (1)

  = m ⊗ R σ R (ε(τ S (a)m ))m

  (1) (2) (0) (1) (1)

  = m ⊗ R σ R (ε(m ))m σ S (a)

  (1) (2) ✭♣♦r (ii)✮ (0) (1)

  = m ⊗ R m σ S (a)

  (0) (1)

  = (m ⊗ R m )a,

  (0) (1) (0) (1) M

  ⊗ R m ) = (m ⊗ R m )a (m) ♦✉ s❡❥❛✱ a(m ✱ ❧♦❣♦ ρ ♣❡rt❡♥❝❡ ❛♦ R C ❝❡♥tr♦ ❞♦ S✲❜✐♠ó❞✉❧♦ M ⊗ ✳ ❆❣♦r❛ ❞❛❞❛ ♦✉tr❛ ❛çã♦ à ❡sq✉❡r❞❛ a⊲m✱ M

  (m) R C ❡♠ M✱ t❛❧ q✉❡ ρ ♣❡rt❡♥❝❡ ❛♦ ❝❡♥tr♦ ❞♦ S✲❜✐♠ó❞✉❧♦ M ⊗ ✱ t❡♠♦s q✉❡

  (0) (1)

  a ⊲ m = a ⊲ (m τ R (ε(m ))

  (0) (1)

  = (a ⊲ m )τ R (ε(m ))

  (0) (1)

  = m τ R (ε(m σ S (a)))

  (0) (1)

  = m τ R (ε(τ S (a)m )) = a · m. ❆❣♦r❛ s❡❥❛♠ M ❡ N C✲❝♦♠ó❞✉❧♦s à ❞✐r❡✐t❛ ❡ f : M −→ N ♠♦r✜s♠♦ ❞❡ C

  ✲❝♦♠ó❞✉❧♦s à ❞✐r❡✐t❛✳ ◆♦t❡ q✉❡ f é R✲❧✐♥❡❛r✳ ▼♦str❡♠♦s ❡♥tã♦ q✉❡ f é S✲❧✐♥❡❛r✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a ∈ S ❡ m ∈ M✱ t❡♠♦s

  (0) (1)

  f (a · m) = f (m τ R (ε(τ S (a)m )))

  (0) (1)

  = f (m )τ R (ε(τ S (a)m ))

  (0) (1)

  = f (m) τ R (ε(τ S (a)f (m) )) = a · f (m). C M

  −→ S R ❊st❡ ❧❡♠❛ ♥♦s ❣❛r❛♥t❡ q✉❡ ❡①✐st❡ ✉♠ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ M ✱ ♣♦✐s t♦❞♦ C✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛ é t❛♠❜é♠ ✉♠ S✲R✲❜✐♠ó❞✉❧♦ ❡ t♦❞♦ ♠♦r✜s♠♦ ❞❡ C✲❝♦♠ó❞✉❧♦s à ❞✐r❡✐t❛ é ✉♠ ♠♦r✜s♠♦ ❞❡ S✲R✲❜✐♠ó❞✉❧♦s✳ ❉❡✜♥✐çã♦ ✷✳✷✻ ❙❡❥❛ B ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❯♠ B✲❝♦♠ó❞✉❧♦ é ✉♠ ❝♦♠ó❞✉❧♦ s♦❜r❡ ♦ R✲❝♦❛♥❡❧ (B, ∆, ε)✳ ❚❡♦r❡♠❛ ✷✳✷✼ ❙❡❥❛ R ✉♠❛ á❧❣❡❜r❛ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k ❡ B B ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❊♥tã♦ ❛ ❝❛t❡❣♦r✐❛ M ❞♦s B✲❝♦♠ó❞✉❧♦s C

  M −→ R R

  á ❞✐r❡✐t❛ é ♠♦♥♦✐❞❛❧✱ t❛❧ q✉❡ ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ M é ❡str✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧✳ B ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r♦✱ ♥♦t❡ q✉❡ R ∈ M ✳ ❉❡ ❢❛t♦✱ ❞❡✜♥✐♠♦s ❛ ❝♦❛çã♦ ❞❡ B ❡♠ R✱ ♣♦r R

  ρ : R −→ R ⊗ R B, r 7−→ 1 R ⊗ s(r). ▼♦str❡♠♦s q✉❡✱ ❞❡ ❢❛t♦✱ é ✉♠❛ ❝♦❛çã♦✳ P❛r❛ t♦❞♦ r ∈ R✱ t❡♠♦s R

  (R ⊗ R ∆)ρ (r) = (R ⊗ R ∆)(1 R ⊗ R s(r)) = 1 R ⊗ R

  1 B ⊗ R s(r), ♣♦r ♦✉tr♦ ❧❛❞♦✱ R R R

  (ρ ⊗ R B)ρ (r) = (ρ ⊗ R B)(1 R ⊗ R s(r)) R = 1 R ⊗ R 1 R ⊗ R s(r). ❆❣♦r❛ ρ é ♠♦r✜s♠♦ ❞❡ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r

  a, r ∈ R ✱ t❡♠♦s R

  ρ (ar) = 1 R ⊗ R s(ar) = 1 R ⊗ R s(a)s(r) = (1 R ⊗ R s(a)) · r R B B = ρ (a) · r. R N ∈ M

  ❙❡❥❛♠ M ❡ N ∈ M ✱ ❡♥tã♦ M ⊗ ✳ ❉❡ ❢❛t♦✱ ❞❡✜♥✐♠♦s ❛ R N ❝♦❛çã♦ ❡♠ M ⊗ ✱ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M ❡ n ∈ N✱ ♣♦r M,N (0) (0) (1) (1) ρ : M ⊗ R N −→ M ⊗ R N ⊗ R B, m ⊗ n 7−→ m ⊗ n ⊗ m n .

  ◆♦t❡ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M ❡ n ∈ N✱ t❡♠♦s M,N M R M R (0) (0) (1) (1) (Id ⊗N ⊗ ∆)ρ (m ⊗ n) =(Id ⊗N ⊗ ∆)(m ⊗ n ⊗ m n )

  (0) (0) (1) (1) (1) (1)

  = m ⊗ n ⊗(m n ) ⊗(m n )

  (1) (2) (0) (0) (1) (1) (1) (1)

  = m ⊗ n ⊗m n ⊗m n

  (1) (1) (2) (2) (0)(0) (0)(0) (0)(1) (0)(1) (1) (1)

  = m ⊗n ⊗m n ⊗m n , ♣♦r ♦✉tr♦ ❧❛❞♦✱ M,N M,N M,N

  (0) (0) (1) (1)

  (ρ ⊗ R B)ρ (m ⊗ n) = (ρ ⊗ R B)(m ⊗ n ⊗ m n )

  (0)(0) (0)(0) (0)(1) (0)(1) (1) (1) M,N = m ⊗n ⊗m n ⊗m n .

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ρ é ♠♦r✜s♠♦ ❞❡ R✲♠ó❞✉❧♦s à ❞✐r❡✐t❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M✱ n ∈ N ❡ r ∈ R✱ t❡♠♦s M,N M,N

  ρ ((m ⊗ n) · r) = ρ (m ⊗ n · r)

  (0) (0) (1) (1)

  = m ⊗ n ⊗ m n s(r)

  (0) (0) (1) (1)

  = (m ⊗ n ⊗ m n ) · r M,N = ρ (m ⊗ n) · r. ❖s ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s sã♦ ♦s ❝❛♥ô♥✐❝♦s✳ ❏á ♠♦str❛♠♦s ♥♦ ❡①❡♠♣❧♦ M,N,P ✶✳✽ q✉❡ a é ♠♦r✜s♠♦ ❞❡ B✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ▼♦str❡♠♦s ❡♥tã♦✱ M M q✉❡ l ❡ r sã♦ ♠♦r✜s♠♦s ❞❡ B✲❝♦♠ó❞✉❧♦s à ❞✐r❡✐t❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M ❡ r ∈ R✱ t❡♠♦s R,M (0) (0) (1) (1)

  (l M ⊗ R B)ρ (r ⊗ m) = (l M ⊗ R B)(r ⊗ m ⊗ r m )

  (0) (1)

  = (l M ⊗ R B)(1 R ⊗ m ⊗ s(r)m )

  (0) (1)

  = m ⊗ s(r)m , ♣♦r ♦✉tr♦ ❧❛❞♦✱ M M

  (ρ ◦ l M )(r ⊗ m) = ρ (r · m) M

  (0) (1)

  = ρ (m τ R (ε(τ S (r)m ))) M

  (0) (1)

  = ρ (m τ R (ε(s(r)m )))

  (0)(0) (0)(1) (1)

  = m ⊗ m τ R (ε(s(r)m ))

  (0) (1) (1)

  = m ⊗ m s(ε(s(r)m ))

  (1) (2) (0) (1) (1)

  = m ⊗ (s(r)m ) s(ε(s(r)m ) ))

  (1) (2)

  

(0) (1)

= m ⊗ s(r)m .

  ❚❛♠❜é♠ t❡♠♦s M,R (0) (0) (1) (1) (r M ⊗ B)ρ (m ⊗ r) = (r M ⊗ B)(m ⊗ r ⊗ m r )

  (0) (1)

  = (r M ⊗ B)(m ⊗ 1 R ⊗ m s(r))

  (0) (1)

  = m ⊗ m s(r), ♣♦r ♦✉tr♦ ❧❛❞♦✱ M M

  (ρ ◦ r M )(m ⊗ r) = ρ (mτ R (r))

  (0) (1)

  = m ⊗ m τ R (r)

  (0) (1) = m ⊗ m s(r).

  P♦rt❛♥t♦✱ ❝♦♥❝❧✉í♠♦s ❛ss✐♠✱ q✉❡ ❛ ❝❛t❡❣♦r✐❛ ❞♦s B✲❝♦♠ó❞✉❧♦s à ❞✐r❡✐t❛ B M

  −→ R R é ♠♦♥♦✐❞❛❧✳ ▼♦str❛r q✉❡ ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ M é ❡s✲ tr✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧ é ❛♥á❧♦❣♦ ❛♦ ❝❛s♦ ❞❛ ❝❛t❡❣♦r✐❛ ❞♦s B✲♠ó❞✉❧♦s à ❞✐r❡✐t❛✳

  

✷✳✻ ❱❡rsõ❡s ❊q✉✐✈❛❧❡♥t❡s ❞❛ ❉❡✜♥✐çã♦ ❞❡ ❇✐✲

❛❧❣❡❜ró✐❞❡

  ◆❡st❛ s❡çã♦ ✐r❡♠♦s ✈❡r ❞✉❛s ✈❡rsõ❡s ❡q✉✐✈❛❧❡♥t❡s ❛ ❞❡✜♥✐çã♦ ❞❡ ❜✐❛❧✲ ❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛✱ ❝♦♠♦ ❞❡✜♥✐❞❛ ♥❡st❡ tr❛❜❛❧❤♦✳ ❆ s❛❜❡r✱ ❡①✐st❡ ✉♠❛ ❢♦r♠✉❧❛çã♦ ♣♦r ✉♠❛ ❛♣❧✐❝❛çã♦ ❝❤❛♠❛❞❛ ❞❡ â♥❝♦r❛ ❡♠ ❬✸✶❪ ❡ ✉♠❛ ❢♦r✲ R ♠✉❧❛çã♦ ♣♦r × ✲❜✐á❧❣❡❜r❛s ❡♠ ❬✷✽❪✳ ▼✐❧✐t❛r✉ ❡ ❇r③❡③✐♥s❦✐ ♠♦str❛r❛♠ ❡♠ ❬✽❪✱ q✉❡ t❛✐s ❢♦r♠✉❧❛çõ❡s sã♦✱ ❞❡ ❢❛t♦✱ ❡q✉✐✈❛❧❡♥t❡s✳ ❉❡✜♥✐çã♦ ✷✳✷✽ ❙❡❥❛♠ R ✉♠❛ á❧❣❡❜r❛ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ e ✉♥✐❞❛❞❡ k ❡ (H, s, t) ✉♠ R ✲❛♥❡❧✳ ❯♠❛ q✉í♥t✉♣❧❛ (H, s, t, ∆, µ) é ✉♠ R

  ✲❜✐❛❧❣❡❜ró✐❞❡ ❝♦♠ â♥❝♦r❛✱ s❡ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿ l (1) ∆ : H −→ H ⊗ R H H

  é ❝♦❛ss♦❝✐❛t✐✈♦✱ ∆(H) ⊆ H × R ❡ ∆ é ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s❀

  (2) µ : H −→ End(R) é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s ❡ ♠♦r✜s♠♦ ❞❡

  R ✲á❧❣❡❜r❛s❀

  (3) P❛r❛ q✉❛✐sq✉❡r r ∈ R ❡ h ∈ H✱ t❡♠♦s

  (i) s(µ(h )(r))h = hs(r)

  (1) (2) ❀

  (ii) t(µ(h )(r))h = ht(r)

  (2) (1) ✳

  ❆ ❛♣❧✐❝❛çã♦ µ é ❝❤❛♠❛❞❛ ❞❡ â♥❝♦r❛✳ e Pr♦♣♦s✐çã♦ ✷✳✷✾ ❙❡❥❛ (H, s, t) ✉♠ R ✲❛♥❡❧✳ ❊♥tã♦✱ (H, s, t, ∆, ε) é ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ (H, s, t, ∆, µ) é ✉♠ R✲ ❜✐❛❧❣❡❜ró✐❞❡ ❝♦♠ â♥❝♦r❛✳ ❉❡♠♦♥str❛çã♦✿ (⇒) ❉❡✜♥❛

  µ : H −→ End(R), t❛❧ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R ❡ h ∈ H✱ t❡♠♦s µ(h)(r) := ε(hs(r)) = ε(ht(r)). ▼♦str❡♠♦s q✉❡ µ é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐q✉❡r

  ′

  r , a ∈ R ✱ r ❡ h ∈ H✱ t❡♠♦s

  

′ ′

  µ(r · h · r )(a) = µ(s(r)t(r )h)(a)

  ′

  = ε(s(r)t(r )hs(a))

  ′

  = rε(hs(a))r

  ′

  = r(µ(h)(a))r

  ′ = (r · µ(h) · r )(a).

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ µ é ♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r h, k ∈ H

  ❡ r ∈ R✱ t❡♠♦s µ(h)(µ(k)(r)) = µ(h)(ε(ks(r)))

  = ε(hs(ε(ks(r)))) = ε(hks(r)) = µ(hk)(r).

  (i) s(µ(h )(r))h = s(ε(h s(r)))h

  (1) (2) (1) (2)

  = s(ε(h t(r)))h

  (1) (2)

  = s(ε(h ))h s(r)

  (1) (2) = hs(r).

  (ii) t(µ(h )(r))h = t(ε(h s(r)))h

  (2) (1) (2) (1)

  = t(ε(h

  (2)

  = s(µ(h

  (1)

  )(1 R ))h

  (2)

  = hs(1 R ) ✭♣♦r (i)✮

  = h, t❛♠❜é♠ t❡♠♦s h

  (1)

  · ε(h

  ) = t(ε(h

  ))h

  (2)

  ))h

  (1)

  = t(µ(h

  (2)

  )(1 R )h

  (1)

  = ht(1 R ) ✭♣♦r (ii)✮ = h.

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ε é ❝❛r❛❝t❡r à ❡sq✉❡r❞❛ ❡♠ (B, s)✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r h, k ∈ H✱ t❡♠♦s ε(hs(ε(k))) = µ(hs(ε(k)))(1 R )

  (2)

  (1)

  (2)

  ′

  ))h

  (1)

  t(r) = ht(r). (⇐)

  ❉❡✜♥❛ ε : H −→ R✱ h 7−→ µ(h)(1 R ) ✳ ▼♦str❡♠♦s q✉❡ ε é ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r, r

  ′

  ∈ R ❡ h ∈ H✱ t❡♠♦s

  ε(r · h · r

  ′

  ) = µ(r · h · r

  )(1 R ) = (r · µ(h) · r

  = s(ε(h

  ′

  )(1 R ) = r(µ(h)(1 R ))r

  ′

  = rε(h)r

  ′ .

  ▼♦str❡♠♦s ❛❣♦r❛ ♦ ❛①✐♦♠❛ ❞❛ ❝♦✉♥✐❞❛❞❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ h ∈ H✱ t❡♠♦s ε(h

  (1)

  ) · h

  (2)

  = µ(h)(µ(s(ε(k)))(1 R )) = µ(h)(µ(ε(k) · 1 H )(1 R )) = µ(h)(ε(k)µ(1 H )(1 R )) = µ(h)(ε(k)) = µ(h)(µ(k)(1 R )) = µ(hk)(1 R ) = ε(hk). R

  P❛r❛ ❞❡✜♥✐♠♦s × ✲❜✐á❧❣❡❜r❛s✱ ♣r❡❝✐s❛♠♦s ❛♥❛❧✐s❛r ❝♦♠ ♠❛✐s ♣r♦✲ ❢✉♥❞✐❞❛❞❡ ♦ ♣r♦❞✉t♦ ❞❡ ❚❛❦❡✉❝❤✐✳ P❛r❛ t❛♥t♦✱ ♥♦t❡ q✉❡✱ s❡ M é ✉♠ e op R

  ✲❜✐♠ó❞✉❧♦✱ ❡♥tã♦ t❛♠❜é♠ é R✲❜✐♠ó❞✉❧♦ ❡ R ✲❜✐♠ó❞✉❧♦✳ ❉❡ ❢❛t♦✱ ❞❡✜♥✐♠♦s ❛s ❡str✉t✉r❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ R ❡ m ∈ M

  ✱ ♣♦r amb := (a ⊗ 1 R ) · m · (b ⊗ 1 R ) ❡ amb := (1 R ⊗ a) · m · (1 R ⊗ b), op

  ❡♠ q✉❡ a ❡ b ❞❡♥♦t❛♠ ❡st❛r❡♠ ❡♠ R ✳ ❆ss✐♠✱ ♣❛r❛ q✉❛✐sq✉❡r M ❡ N e ❞♦✐s R ✲❜✐♠ó❞✉❧♦s✱ ❞❡✜♥✐♠♦s ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧

  Z a a M ⊗ k a N := (M ⊗ k N )/F = M ⊗ R N, ❡♠ q✉❡ F = span{am ⊗ n − m ⊗ an | a ∈ R}✳ ❆❣♦r❛ ❞❡✜♥✐♠♦s a Z

  X X

  X M a ⊗ k N a := { m i ⊗n i ∈ M ⊗ k N | m i a⊗n i = m i ⊗n i a}. ❆ss✐♠✱ ❞❡✜♥✐♠♦s ♦ ♣r♦❞✉t♦ ❞❡ ❚❛❦❡✉❝❤✐✱ ♣♦r b

  Z Z M × R N := a M ⊗ k a N b , e a b q✉❡ ♣♦ss✉✐ ❡str✉t✉r❛ ❞❡ R ✲❜✐♠ó❞✉❧♦ ❞❛❞❛ ♣♦r

  X X (a ⊗ b)( m i ⊗ n i )(c ⊗ d) = am i c ⊗ bn i i i e P

  d, m i ⊗ n i ∈ M × R N ♣❛r❛ q✉❛✐sq✉❡r a ⊗ b, c ⊗ d ∈ R ❡ ✳ ❆❣♦r❛✱ P❛r❛ e i M, N

  ❡ P três R ✲❜✐♠ó❞✉❧♦s✱ t❡♠♦s b,d Z Z M × R N × R P = a M ⊗ k ac N ⊗ k c P d . a,c b bd

  ❖r✐❣✐♥❛❧♠❡♥t❡✱ ❛ ❢♦r♠✉❧❛çã♦ ❞❡ ❜✐❛❧❣❡❜ró✐❞❡ ❢♦✐ ❢❡✐t❛ ✉s❛♥❞♦ ♦ ♣r♦❞✉t♦ ❚❛❦❡✉❝❤✐ ❞❡ ♠❛♥❡✐r❛ ♠❛✐s ❞✐r❡t❛ ❡♠ ❬✷✽❪✳ ❖ ♣r♦❜❧❡♠❛ é q✉❡ ❡❧❡ ♥ã♦ é ❛ss♦❝✐❛t✐✈♦✳ P❛r❛ ❝♦♥t♦r♥❛r ❡ss❡ ♣r♦❜❧❡♠❛✱ ❞❡✜♥✐♠♦s ❞✉❛s ✐♥❝❧✉sõ❡s ❝❛♥ô♥✐❝❛s

  α : (M × R N ) × R P −→ M × R N × R P ❡

  ′ α : M × R (N × R P ) −→ M × R N × R P. e

  ▲❡♠❛ ✷✳✸✵ ❙❡❥❛ M ✉♠ R ✲❜✐♠ó❞✉❧♦✳ ❊♥tã♦✱ ❡①✐st❡♠ ♠♦r✜s♠♦s ❞❡ e R

  ✲❜✐♠ó❞✉❧♦s θ : M × R End k (R) −→ M

  P P m i ⊗ f i 7−→ f i (1 R )m i ❡

  ′

  θ : End k (R) × R M −→ M P P f i ⊗ m i 7−→ f i (1 R )m i . k e

  (R) ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r♦✱ ♥♦t❡ q✉❡ End é ✉♠ R ✲❜✐♠ó❞✉❧♦✳ ❉❡ e

  −→ End k (R) ❢❛t♦✱ ❞❡✜♥❛ i : R ✱ t❛❧ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r a, b✱ r ∈ R k (R) ❡ f ∈ End ✱ i(a ⊗ b)(r) = arb✳ P♦rt❛♥t♦✱ ❞❡✜♥✐♠♦s ♣❛r❛ q✉❛✐s✲ k (R) e q✉❡r a, b, c, d✱ r ∈ R ❡ f ∈ End ❛ ❡str✉t✉r❛ ❞❡ R ✲❜✐♠ó❞✉❧♦s ❡♠ End k (R)

  ✱ ♣♦r ((a ⊗ b) · f · (c ⊗ d))(r) = i(a ⊗ b)(f (i(c ⊗ d)(r))) = a(f (crd))b. ❆❣♦r❛ ❞❡✜♥❛ e

  θ : M × End k (R) −→ M (m, f ) 7−→ f (1 R )m, q✉❡ é k✲❧✐♥❡❛r✳ P♦rt❛♥t♦✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛

  θ : b M ⊗ k End k (R) −→ M R m ⊗ f 7−→ f (1 )m. ◆♦t❡ q✉❡

  θ(am ⊗ f − m ⊗ a · f ) = b b θ(am ⊗ f ) − b θ(m ⊗ a · f ) = f (1 R )am − a · f (1 R )m = a · f (1 R )m − a · f (1 R )m = 0,

  ♦✉ s❡❥❛✱ F ⊆ ker(bθ)✳ P♦rt❛♥t♦✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ θ : M ⊗ R End k (R) −→ M m ⊗ f 7−→ f (1 R )m.

  P♦rt❛♥t♦✱ ❜❛st❛ ❞❡✜♥✐r♠♦s θ := θ| M × End (R) ✳ ▼♦str❡♠♦s q✉❡ θ é e R e ♠♦r✜s♠♦ ❞❡ R ✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a ⊗ b, c ⊗ d ∈ R ❡ P i ✱ t❡♠♦s m i ⊗ f i ∈ M × R End k (R)

X X

  θ((a ⊗ b)( m i ⊗ f i )(c ⊗ d)) = θ( am i c ⊗ bf i i i

  d) X = bf i d(1 R )am i c i

  X = f i (d)b am i c i

  X = a b f i (d) m i c i

  X = (a ⊗ b)f i d(1 R )m i c i

  X = (a ⊗ b) · f i (1 R )m i dc i

  X = (a ⊗ b) · ( f i (1 R )m i ) · (c ⊗ d) i

  X = (a ⊗ b) · θ( m i ⊗ f i ) · (c ⊗ d). i

  ′

  P❛r❛ θ é ❛♥á❧♦❣♦✳ e R ❉❡✜♥✐çã♦ ✷✳✸✶ ❙❡❥❛ (H, s, t) ✉♠ R ✲❛♥❡❧✳ ❯♠❛ × ✲❝♦á❧❣❡❜r❛ é ✉♠❛ tr✐♣❧❛ (H, ∆, µ) ❡♠ q✉❡✿

  (1) ∆ : H −→ H × R H k (R) e ❡ µ : H −→ End sã♦ ♠♦r✜s♠♦s ❞❡ R

  ✲❜✐♠ó❞✉❧♦s❀

  ′

  (2) (i) α ◦ (∆ × R

  H) ◦ ∆ = α ◦ (H × R ∆) ◦ ∆ ✭❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡✮❀

  ′

  (ii) θ ◦ (H × R µ) ◦ ∆ = Id H = θ ◦ (µ × R

  H) ◦ ∆ ✭❝♦✉♥✐t❛❧✐❞❛❞❡✮✳

  ∆ R é ❝❤❛♠❛❞♦ ❝♦♠✉❧t✐♣❧✐❝❛çã♦ ❡ µ é ❝❤❛♠❛❞♦ ❝♦✉♥✐❞❛❞❡ ❞❛ × ✲❝♦á❧❣❡❜r❛✳ R H −→ H ⊗ R H

  Pr♦♣♦s✐çã♦ ✷✳✸✷ ❙❡❥❛♠ i : H × ❛ ✐♥❝❧✉sã♦ ❝❛♥ô✲ R H k (R) ♥✐❝❛✱ ∆ : H −→ H × ❡ µ : H −→ End ❞♦✐s ♠♦r✜s♠♦s ❞❡ e R R

  ✲❜✐♠ó❞✉❧♦s✳ ❊♥tã♦ (H, ∆, µ) é ✉♠❛ × ✲❝♦á❧❣❡❜r❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ (H, ∆, ε)

  é ✉♠ R✲❝♦❛♥❡❧✱ ❝♦♠ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦ ❞❛❞❛✱ ♣❛r❛ q✉❛✐s✲

  ′ ′ ′

  ∈ R = (r ⊗ r ) · h q✉❡r h ∈ H ❡ r, r ✱ ♣♦r r ⊲ h ⊳ r ✱ ❡♠ q✉❡ ∆ = i ◦ ∆ ❡ ε(h) = µ(h)(1 R )

  ✳ R ❉❡♠♦♥str❛çã♦✿ ❙❡ (H, ∆, µ) é ✉♠❛ × ✲❝♦á❧❣❡❜r❛✱ ❡♥tã♦ ∆ é ❛✉✲ t♦♠❛t✐❝❛♠❡♥t❡ ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❉❡✈❡♠♦s ♠♦str❛r q✉❡ ε é

  ′

  ∈ R ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✱ ❞❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r, r ❡ h ∈ H✱ t❡♠♦s

  

′ ′

  ε(r ⊲ h ⊳ r ) = µ(r ⊲ h ⊳ r )(1 R )

  ′

  = µ((r ⊗ r ) · h)(1 R )

  ′

  = ((r ⊗ r ) · µ(h))(1 R )

  ′

  = rµ(h)(1 R )r

  ′ = rε(h)r .

  2

  : H × R H × R H −→ H ⊗ R H ⊗ R H P❛r❛ ❛ ❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡✱ ❞❡✜♥❛ i ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛✱ ❞❡ss❛ ❢♦r♠❛✱ t❡♠♦s

  (∆ ⊗ R

  H) ◦ ∆ = ((i ◦ ∆) ⊗ R

  H) ◦ i ◦ ∆ = (i ⊗ R H)(∆ ⊗ R

  H) ◦ i ◦ ∆

  2

  = i ◦ α ◦ (∆ × R

  H) ◦ ∆

  

2 ′

  = i ◦ α ◦ (H × R ∆) ◦ ∆ = (H ⊗ R i)(H ⊗ R ∆) ◦ i ◦ ∆ = (H ⊗ R (i ◦ ∆)) ◦ i ◦ ∆ = (H ⊗ R ∆) ◦ ∆.

  ▼♦str❡♠♦s ❛ ❝♦✉♥✐t❛❧✐❞❛❞❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ h ∈ H✱ t❡♠♦s (ε ⊗ R

  H) ◦ ∆(h) = ε(h ) ⊲ h

  (1) (2)

  = µ(h )(1 R ) ⊲ h

  (1) (2)

  = θ (µ(h ) ⊗ R h )

  (1) (2)

  = θ ◦ (µ × R H)∆(h) = h, t❛♠❜é♠ t❡♠♦s (H ⊗ R ε) ◦ ∆(h) = h ⊳ ε(h )

  (1) (2)

  = ε(h ) · h

  (2) (1)

  = µ(h )(1 R ) · h

  (2) (1)

  = θ(h ⊗ R µ(h ))

  (1) (2)

  = θ ◦ (H × R µ) ◦ ∆(h) = h. ❊q✉✐✈❛❧❡♥t❡♠❡♥t❡ s❡ ∆ é ❝♦❛ss♦❝✐❛t✐✈♦ ❡ ✈❛❧❡ ❛ ❝♦✉♥✐t❛❧✐❞❛❞❡ ♣❛r❛ ε✱ ❡♥tã♦ ♠♦str❛✲s❡ q✉❡ ∆ é ❝♦❛ss♦❝✐❛t✐✈♦ ❡ ✈❛❧❡ ❛ ❝♦✉♥✐t❛❧✐❞❛❞❡ ♣❛r❛ µ✳ e ❉❡✜♥✐çã♦ ✷✳✸✸ ❙❡❥❛ (H, s, t) ✉♠ R ✲❛♥❡❧✳ ❆ tr✐♣❧❛ (H, ∆, µ) é ✉♠❛ × R R

  ✲❜✐á❧❣❡❜r❛✱ s❡ é ✉♠❛ × ✲❝♦á❧❣❡❜r❛✱ t❛❧ q✉❡ ∆ ❡ µ sã♦ ♠♦r✜s♠♦s e ❞❡ R ✲❛♥é✐s✳ e ❚❡♦r❡♠❛ ✷✳✸✹ ❙❡❥❛ H ✉♠ R ✲❛♥❡❧✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥✲ ❝✐❛ 1 ❛ 1 ❡♥tr❡ ❡str✉t✉r❛s ❞❡ R✲❜✐❛❧❣❡❜ró✐❞❡s à ❡sq✉❡r❞❛ ❡♠ H ❡ ❡str✉✲ R t✉r❛s ❞❡ × ✲❜✐á❧❣❡❜r❛s ❡♠ H✳

  ❉❡♠♦♥str❛çã♦✿ (⇒) ❙❡❥❛ (H, ∆, ε) ✉♠❛ ❡str✉t✉r❛ ❞❡ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛✳ ❉❡✜♥❛ ∆ : H −→ H × R H ❝♦♠♦ ❛ ❝♦✲r❡str✐çã♦ ❞❡ ∆✳ ❏á t❡♠♦s q✉❡ ∆ é ♠♦r✜s♠♦ ❞❡ R✲á❧❣❡❜r❛s✳ ❋❛❧t❛ ♠♦str❛r♠♦s q✉❡ ∆◦I H =

  1 R · b = s(a) ⊗ R t(b) = I H

  (a ⊗ b)(c), t❛♠❜é♠ t❡♠♦s µ ε (hk)(a) = ε(hks(a))

  (R)

  = ε(s(a)t(b)s(c)) = ε(s(a)s(c)t(b)) = ε(s(ac)t(b)) = ε(s(ac)s(ε(t(b)))) = ε(s(ac)s(b)) = ε(s(acb)) = acb = I End k

  µ ε : H −→ End k (R), h 7−→ µ ε (h), t❛❧ q✉❡ µ ε (h)(r) = ε(hs(r)) ✱ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R ❡ h ∈ H✳ ❆ss✐♠✱ t❡♠♦s µ ε (I H (a ⊗ b))(c) = µ ε (s(a)t(b))(c)

  I End k (R) (a ⊗ b)(c) = acb, ♣❛r❛ q✉❛✐sq✉❡r a, b ❡ c ∈ R✳ P♦rt❛♥t♦✱ ❞❡✜♥❛

  ❆❣♦r❛ ♥♦t❡ q✉❡✱ ❛ ✉♥✐❞❛❞❡ ❞♦ R e ✲❛♥❡❧ End k (R) é ❞❡✜♥✐❞❛ ♣♦r

  × R H (a ⊗ b).

  = ∆(a · 1 H · b) = a · ∆(1 H ) · b = a · 1 H ⊗ R

  I H

  ✳ ❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s ∆(I H (a ⊗ b)) = ∆(s(a)t(b))

  × R H (a ⊗ b) = s(a) ⊗ t(b)

  I H

  −→ H × R H ✱ t❛❧ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ R✱

  × R H : R e

  ✱ t❛❧ q✉❡ I H (a ⊗ b) = s(a)t(b) ✱ ❡ I H

  ✱ ❡♠ q✉❡ I H é ❛ ✉♥✐❞❛❞❡ I H : R e −→ H

  × R H

  = ε(hs(ε(ks(a)))) = µ ε (h)(ε(ks(a)))

  = µ ε (h)(µ ε (k)(a)) = µ ε (h) ◦ µ ε (k)(a),

  ♣❛r❛ q✉❛✐sq✉❡r a, b, c ∈ R ❡ h, k ∈ H✳ ❆❣♦r❛ ♥♦t❡ q✉❡✱

  2

  i ◦ α ◦ (∆ × R

  H) ◦ ∆ = (i ⊗ R H)(∆ ⊗ R

  H) ◦ i ◦ ∆ = ((i ◦ ∆) ⊗ R

  H) ◦ i ◦ ∆ = (∆ ⊗ R

  H) ◦ ∆ = (H ⊗ R ∆) ◦ ∆ = (H ⊗ R (i ◦ ∆)) ◦ i ◦ ∆ = (H ⊗ R i)(H ⊗ R ∆) ◦ i ◦ ∆

  2 ′

  = i ◦ α ◦ (H × R ∆) ◦ ∆, R ′ H) ◦ ∆ = α ◦ (H × R ∆) ◦ ∆ s❡❣✉❡ q✉❡ α ◦ (∆ × ✳ ❆❣♦r❛✱ ♣❛r❛ t♦❞♦ h ∈ H

  ✱ t❡♠♦s θ ◦ (H × R µ ε ) ◦ ∆(h) = θ(h × R µ ε (h ))

  (1) (2)

  = µ ε (h )(1 R ) · h

  (2) (1)

  = h ⊳ µ ε (h )(1 R )

  (1) (2)

  = t(µ ε (h )(1 R ))h

  (2) (1)

  = t(ε(h s(1 R )))h

  (2) (1)

  = t(ε(h ))h

  (2) (1)

  = h ⊳ ε(h ) = h,

  (1) (2)

  t❛♠❜é♠ t❡♠♦s

  ′ ′

  θ ◦ (µ ε × R

  H) ◦ ∆(h) = θ (µ ε (h ) × R h )

  (1) (2)

  = (µ ε (h )(1 R ))h

  (1) (2)

  = s(ε(h s(1 R )))h

  (1) (2)

  = s(ε(h ))h

  (1) (2)

  = ε(h ) ⊲ h

  (1) (2) = h.

  (⇐) R ❈♦♥s✐❞❡r❡ (H, ∆, µ) ✉♠❛ × ✲❜✐á❧❣❡❜r❛✳ ❉❡✜♥❛ R H −→ H ⊗ R H ∆ := i ◦ ∆ : H −→ H ⊗ R

  H, ❡♠ q✉❡ i : H × é ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛✳ ❉❡✜♥❛ t❛♠❜é♠ R ) ❛ ❝♦✉♥✐❞❛❞❡ ε : H −→ R ❝♦♠♦ ε(h) = µ(h)(1 ✳ ◆♦✈❛♠❡♥t❡✱ s❡ ∆ ❡ e

  µ sã♦ ♠♦r✜s♠♦s ❞❡ R ✲❜✐♠ó❞✉❧♦s ❡ ♠♦r✜s♠♦s ❞❡ á❧❣❡❜r❛s✱ ❡♥tã♦ ∆ é ♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❡ ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✱ ❡ ε é ♠♦r✜s♠♦ ❞❡ R

  ✲❜✐♠ó❞✉❧♦✳ ◗✉❡ ε é ❝❛r❛❝t❡r à ❡sq✉❡r❞❛ ❡♠ (H, s)✱ ❥á ❢♦✐ ❢❡✐t♦ ♥❛ ♣r♦♣♦s✐çã♦ ✷✳✷✾✳

  ❈❛♣ít✉❧♦ ✸ ❍♦♣❢ ❆❧❣❡❜ró✐❞❡s

  ❙❛❜❡♠♦s q✉❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢ é ✉♠❛ ❜✐á❧❣❡❜r❛ H ❡q✉✐♣❛❞❛ ❝♦♠ ✉♠❛ ❛♣❧✐❝❛çã♦ S : H −→ H ❞❡♥♦♠✐♥❛❞❛ ❛♥tí♣♦❞❛✳ ❆ ♥♦çã♦ ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡st❛ ❡str✉t✉r❛✳ ❊①✐st❡♠ ♦✉tr❛s ♥♦çõ❡s q✉❡ ❣❡♥❡r❛❧✐③❛♠ á❧❣❡❜r❛s ❞❡ ❍♦♣❢✱ ❝♦♠♦ ✈❛♠♦s ✈❡r ♥♦ ✜♥❛❧ ❞❡st❡ ❝❛♣ít✉❧♦✱ ♠❛s ❛♦ ❝♦♥trár✐♦ ❞❡ ❜✐❛❧❣❡❜ró✐❞❡s✱ t❛✐s ♥♦çõ❡s ♥ã♦ sã♦ ❡q✉✐✈❛❧❡♥t❡s✳

  ❆ ❛♥tí♣♦❞❛ S✱ ❝♦♠♦ ♠❡♥❝✐♦♥❛❞❛ ❛❝✐♠❛✱ é ✉♠ ♠♦r✜s♠♦ ❞❡ ❜✐á❧❣❡✲ op ❜r❛s ❞❡ H ♣❛r❛ H cop ✳ ▼❛s✱ ♥ã♦ ♣❛r❡❝❡ s❡r ♣♦ssí✈❡❧ ❞❡✜♥✐r ✉♠ ❍♦♣❢ á❧❣❡❜ró✐❞❡ ❜❛s❡❛❞♦ ♥❡st❛ ❛♥❛❧♦❣✐❛✳ P♦✐s s❡ H é ✉♠ ❜✐❛❧❣❡❜ró✐❞❡ à ❡s✲ cop op q✉❡r❞❛✱ s❡✉ ♦♣♦st♦ ❝♦✲♦♣♦st♦ H é ✉♠ ❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛ ❡ ♥ã♦ op

  ❡①✐st❡ ✉♠❛ ♥♦çã♦ s❡♥s❛t❛ ❞❡ ♠♦r✜s♠♦ ❞❡ ❜✐❛❧❣❡❜ró✐❞❡s H −→ H cop ✳ P♦rt❛♥t♦✱ s❡ q✉✐s❡r♠♦s q✉❡ ❛ ❛♥tí♣♦❞❛ ❞❡ ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✱ s❡❥❛ ✉♠ op op ♠♦r✜s♠♦ ❞❡ ❜✐❛❧❣❡❜ró✐❞❡s H −→ H cop ✱ ♣r❡❝✐s❛♠♦s q✉❡ H ❡ H cop t❡✲ ♥❤❛♠ ❛ ♠❡s♠❛ ❡str✉t✉r❛ ❞❡ ❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ ❛ á❧❣❡❜r❛ s✉❜❥❛❝❡♥t❡ H ♣r❡❝✐s❛ s❡r t❛♥t♦ ✉♠ ❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛✱ q✉❛♥t♦ ✉♠ ❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳

  ✸✳✶ ❉❡✜♥✐çã♦ ❡ ❊①❡♠♣❧♦s

  ❉❡✜♥✐çã♦ ✸✳✶ ❙❡❥❛♠ R ❡ L ❞✉❛s á❧❣❡❜r❛s s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k ✳ ❯♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ s♦❜r❡ ❛s á❧❣❡❜r❛s ❞❡ ❜❛s❡ R ❡ L é ✉♠❛ tr✐♣❧❛

  H = (H L , H R , S) L R

  ✱ ❡♠ q✉❡ H é ✉♠ L✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛ ❡ H é ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✱ s♦❜r❡ ❛ ♠❡s♠❛ k✲á❧❣❡❜r❛ s✉❜❥❛❝❡♥t❡ H✳ ❆ ❛♥tí♣♦❞❛ é ✉♠ ♠♦r✜s♠♦ ❞❡ k✲♠ó❞✉❧♦ S : H −→ H✳ ❉❡♥♦t❡ ❛s ❡str✉✲ e R R , t R ) t✉r❛s ❞❡ R ✲❛♥❡❧ ❡ ❞❡ R✲❝♦❛♥❡❧ ❞❡ H ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r (H, s R , ε R ) e

  ❡ (H, ∆ ✳ ❙✐♠✐❧❛r♠❡♥t❡✱ ❞❡♥♦t❡ ❛s ❡str✉t✉r❛s ❞❡ L ✲❛♥❡❧ ❡ ❞❡ L✲

  L L , t L ) L , ε L )

  ❝♦❛♥❡❧ ❞❡ H ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r (H, s ❡ (H, ∆ ✳ ❉❡♥♦t❡ R ) R L ) L ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦ R✲❛♥❡❧ (H, s ♣♦r µ ❡ ❞♦ L✲❛♥❡❧ (H, s ♣♦r µ ✳ ❊st❛s ❡str✉t✉r❛s ❡stã♦ s✉❥❡✐t❛s ❛♦s s❡❣✉✐♥t❡s ❛①✐♦♠❛s ❞❡ ❝♦♠♣❛t✐❜✐❧✐✲ ❞❛❞❡✿

  (i) s L ◦ ε L ◦ t R = t R R ◦ ε R ◦ t L = t L L ◦ ε L ◦ s R = s R ✱ s ✱ t ❡ t R ◦ ε R ◦ s L = s L

  ❀ (ii) (∆ L ⊗ R

  H) ◦ ∆ R = (H ⊗ L ∆ R ) ◦ ∆ L ❡

  (∆ R ⊗ L

  H) ◦ ∆ L = (H ⊗ R ∆ L ) ◦ ∆ R ❀

  (iii) S(t L (l)ht R (r)) = s R (r)S(h)s L (l) ✱ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R✱ l ∈ L ❡ h ∈ H

  ❀ (iv) µ L ◦ (S ⊗ L

  H) ◦ ∆ L = s R ◦ ε R ❡

  µ R ◦ (H ⊗ R S) ◦ ∆ R = s L ◦ ε L ✳

  ❖❜s❡r✈❛çã♦ ✸✳✷ (1) P❡❧♦s ❛①✐♦♠❛s ❞❡ ❜✐❛❧❣❡❜ró✐❞❡s✱ t♦❞♦s ♦s ♠♦r✲ L ◦ε L L ◦ε L R ◦ε R R ◦ε R ✜s♠♦s s ✱ t ✱ s ❡ t sã♦ ✐❞❡♠♣♦t❡♥t❡s H −→ H✳ L R P♦rt❛♥t♦✱ ♦ ❛①✐♦♠❛ (i) ♥♦s ❞✐③ q✉❡ ❛s ✐♠❛❣❡♥s ❞❡ s ❡ t ✱ t❛♠❜é♠ R L ❛s ✐♠❛❣❡♥s ❞❡ s ❡ t ✱ sã♦ s✉❜á❧❣❡❜r❛s ❝♦✐♥❝✐❞❡♥t❡s ❞❡ H✳ ■st♦ L ✐♠♣❧✐❝❛ q✉❡ ∆ ♥ã♦ é ❛♣❡♥❛s ♠♦r✜s♠♦ ❞❡ L✲❜✐♠ó❞✉❧♦s✱ ♠❛s t❛♠✲ R ❜é♠ ♠♦r✜s♠♦ ❞❡ R✲❜✐♠ó❞✉❧♦s✳ ❙✐♠❡tr✐❝❛♠❡♥t❡✱ ∆ é ♠♦r✜s♠♦ ❞❡ L✲❜✐♠ó❞✉❧♦s✳ ❉❡ss❛ ❢♦r♠❛✱ ♦ ❛①✐♦♠❛ (ii) ❢❛③ s❡♥t✐❞♦✳

  (2) ❖ k✲♠ó❞✉❧♦ s✉❜❥❛❝❡♥t❡ H ❞❡ ✉♠ ❜✐❛❧❣❡❜ró✐❞❡ ✭à ❡sq✉❡r❞❛ ♦✉ à ❞✐r❡✐t❛✮ é ✉♠ ❝♦♠ó❞✉❧♦ à ❡sq✉❡r❞❛ ❡ à ❞✐r❡✐t❛✱ ✈✐❛ ♦ ❝♦♣r♦❞✉t♦✳ P♦rt❛♥t♦✱ ♦ k✲♠ó❞✉❧♦ s✉❜❥❛❝❡♥t❡ H ❞❡ ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✱ é ✉♠ H L R

  ✲❝♦♠ó❞✉❧♦ à ❡sq✉❡r❞❛ ❡ ✉♠ H ✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛✱ ✈✐❛ ❛♦s L R ❝♦♣r♦❞✉t♦s ∆ ❡ ∆ ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖ ❛①✐♦♠❛ (ii) ❡①♣r❡ss❛ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ q✉❡ ❛s ❝♦❛çõ❡s r❡❣✉❧❛r❡s ❝♦♠✉t❛♠✱ ♦✉ s❡❥❛✱ H é L R R L ✉♠ H ✲H ✲❜✐❝♦♠ó❞✉❧♦ ❡ t❛♠❜é♠ ✉♠ H ✲H ✲❜✐❝♦♠ó❞✉❧♦✳ ❆❧t❡r♥❛t✐✈❛♠❡♥t❡✱ ❛ ♣r✐♠❡✐r❛ ✐❞❡♥t✐❞❛❞❡ ❞♦ ❛①✐♦♠❛ (ii)✱ ♥♦s ❞✐③ L R q✉❡ ∆ é ✉♠ ♠♦r✜s♠♦ ❞❡ H ✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❙✐♠❡tr✐✲ R L

  ❝❛♠❡♥t❡✱ ∆ é ✉♠ ♠♦r✜s♠♦ ❞❡ H ✲❝♦♠ó❞✉❧♦ à ❡sq✉❡r❞❛✳ ❆ L s❡❣✉♥❞❛ ✐❞❡♥t✐❞❛❞❡ ❞♦ ❛①✐♦♠❛ (ii)✱ ♣♦❞❡ s❡r ❧✐❞❛ ❝♦♠♦ ❛ H ✲ R R ❝♦❧✐♥❡❛r✐❞❛❞❡ à ❞✐r❡✐t❛ ❞❡ ∆ ♦✉ ❛ H ✲❝♦❧✐♥❡❛r✐❞❛❞❡ à ❡sq✉❡r❞❛ L ❞❡ ∆ ✳

  (3) ❖ ❛①✐♦♠❛ (iii) ❢♦r♠✉❧❛ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❛♥tí♣♦❞❛ s❡r ♠♦r✜s♠♦ ❞❡ R

  ✲L✲❜✐♠ó❞✉❧♦s✱ ♥❡❝❡ssár✐❛ ♣❛r❛ q✉❡ ♦ ❛①✐♦♠❛ (iv) ❢❛ç❛ s❡♥t✐❞♦❀ (4)

  ❆♥á❧♦❣♦ ❛♦s ❛①✐♦♠❛s ❞❡ á❧❣❡❜r❛s ❞❡ ❍♦♣❢✱ ♦ ❛①✐♦♠❛ (iv) ♥♦s ❞✐③ q✉❡ ❛ ❛♥tí♣♦❞❛ é ❛ ✐♥✈❡rs❛ ❞❛ ✐❞❡♥t✐❞❛❞❡ ♣❡❧♦ ♣r♦❞✉t♦ ❞❡ ❝♦♥✲ ✈♦❧✉çã♦✱ ❡♠ ❛❧❣✉♠ s❡♥t✐❞♦ ❣❡♥❡r❛❧✐③❛❞♦✳ ❆ ♥♦çã♦ ❞❡ ♣r♦❞✉t♦ ❞❡

  ❝♦♥✈♦❧✉çã♦ ♥♦ ❝❛s♦ ❞❡ ❞✉❛s á❧❣❡❜r❛s ❜❛s❡s ❞✐st✐♥t❛s L ❡ R✱ é ♠❛✐s ❝♦♠♣❧✐❝❛❞♦ ❡ ♥ã♦ ❡stá ♥♦ ❡s❝♦♣♦ ❞❡st❡ tr❛❜❛❧❤♦✳

  ❉❛ ❉❡✜♥✐çã♦ ✸✳✶ t❡♠♦s q✉❡ ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ H ♣♦ss✉✐ ❞✉❛s ❡s✲ L R tr✉t✉r❛s ❞❡ ❜✐❛❧❣❡❜ró✐❞❡s H ❡ H ❡ ❞❡ss❛ ❢♦r♠❛✱ ✉s❛♠♦s ❞✉❛s ✈❡rsõ❡s L R ❞❛ ♥♦t❛çã♦ ❞❡ ❙✇❡❡❞❧❡r ♣❛r❛ ♦s ❝♦♣r♦❞✉t♦s ∆ ❡ ∆ ✳ P❛r❛ t♦❞♦ h ∈ H✱ L (h) = h ⊗ L h R (h) = h ⊗ R h (1) (2)

  ✉s❛r❡♠♦s ∆ (1) (2) ❡ ∆ ✱ ✜❝❛♥❞♦ s✉❜❡♥✲ t❡♥❞✐❞♦ ♦ s♦♠❛tór✐♦✳ ❊①❡♠♣❧♦ ✸✳✸ ✭➪❧❣❡❜r❛s ❞❡ ❍♦♣❢✮ ❙❡❥❛ H ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k✳ ❖❜✈✐❛♠❡♥t❡ H é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ s♦❜r❡ á❧✲ L R ❣❡❜r❛s ❜❛s❡ R = k = L✳ ❆♠❜♦s ♦s ❜✐❛❧❣❡❜ró✐❞❡s H ❡ H sã♦ ✐❣✉❛✐s ❛ k

  ✲❜✐á❧❣❡❜r❛ H ❡ ❛ ❛♥tí♣♦❞❛ S ❞❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢ H✱ s❛t✐s❢❛③ ♦s ❛①✐♦♠❛s ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳ ❊①❡♠♣❧♦ ✸✳✹ ✭➪❧❣❡❜r❛s ❞❡ ❍♦♣❢ ❢r❛❝❛s✮ ❯♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢ ❢r❛❝❛ é ✉♠❛ ❜✐á❧❣❡❜r❛ ❢r❛❝❛ H ❡q✉✐♣❛❞❛ ❝♦♠ ✉♠ ♠♦r✜s♠♦ k✲❧✐♥❡❛r S : H −→ H

  ✱ s✉❥❡✐t♦ ❛♦s s❡❣✉✐♥t❡s ❛①✐♦♠❛s ❞❡ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡✱ P❛r❛ t♦❞♦ h ∈ H✱ L (i) h S(h ) = ⊓ (h)

  (1) (2) ❀ R

  (ii) S(h )h = ⊓ (h)

  (1) (2) ❀

  (iii) S(h )h S(h ) = S(h)

  (1) (2) (3) ✱ L R

  ❡ q✉❡ ❛s ❛♣❧✐❝❛çõ❡s ⊓ ❡ ⊓ ❢♦r❛♠ ❞❡✜♥✐❞❛s ♥♦ ❡①❡♠♣❧♦ ✷✳✶✶✳ ❉❡✜♥❛ R L e R := Im(⊓ ) )

  ❡ L := Im(⊓ ✱ ❞❡ss❛ ❢♦r♠❛✱ ❛s ❡str✉t✉r❛s ❞❡ R ✲❛♥❡❧ R R , t R ) ❡ R✲❝♦❛♥❡❧ ❞♦ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛ H ✱ sã♦ ❞❛❞❛s ♣♦r (H, s R , ε R ) R (r) = r R (r) = ❡ (H, ∆ ✱ ❡♠ q✉❡✱ ♣❛r❛ t♦❞♦ r ∈ R✱ t❡♠♦s s ✱ t R ε(r1 )1 R := π R ◦ ∆ R := ⊓ R : H ⊗ k

  (1) (2) ✱ t❛♠❜é♠✱ ∆ ❡ ε ✱ ❡♠ q✉❡ π e

  H −→ H ⊗ R H ✱ é ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛✳ ❆s ❡str✉t✉r❛s ❞❡ L ✲❛♥❡❧ ❡

  L L L , t L ) ✲❝♦❛♥❡❧ ❞♦ L✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛ H ✱ sã♦ ❞❛❞❛s ♣♦r (H, s L , ε L ) L (l) = l L (l) =

  ❡ (H, ∆ ✱ ❡♠ q✉❡✱ ♣❛r❛ t♦❞♦ l ∈ L✱ t❡♠♦s s ✱ t L ε(1 l)1 L := π L ◦∆ L := ⊓ L : H ⊗ k H −→

  (2) (1) ✱ t❛♠❜é♠✱ ∆ ❡ ε ✱ ❡♠ q✉❡ π

  H ⊗ L H ✱ é ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛✳ ❊st❛s ❡str✉t✉r❛s ❥✉♥t♦ ❝♦♠ ❛ ❛♥tí♣♦❞❛

  S ❞❡ H✱ ❝♦♥st✐t✉❡♠ ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳

  ❉❡ ❢❛t♦✱ ♠♦str❡♠♦s ❛s ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s✳ ❆♥t❡s✱ ❧✐st❛♠♦s ❛❧❣✉✲ ♠❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ✈❛♠♦s ✉s❛r✱ ❛❧é♠ ❞❛ q✉❡ ❥á s❛❜❡♠♦s✿

  (1) ∆(1 H ) ∈ R ⊗ L L ❀ (2) ⊓ (h) = ε(S(h)1 )1 = S(1 )ε(1

  h) R

(1) (2) (1) (2) ❀

(3) ⊓ (h) = 1 ε(1 S(h)) = ε(h1 )S(1 )

  (1) (2) (1) (2) ✱ ♣❛r❛ t♦❞♦ h ∈ H✳

  ❆s ♣r♦♣r✐❡❞❛❞❡s (2) ❡ (3) ❡stã♦ ♣r♦✈❛❞❛s ❡♠ ❬✷✷❪✳ ▼♦str❡♠♦s ❛ ♣r♦♣r✐✲ ❡❞❛❞❡ (1)✳ ❉❡ ❢❛t♦✱ L L

  (H ⊗ ⊓ )∆(1 H ) = 1 ⊗ ⊓ (1 )

  (1) (2) ′ ′

  = 1 ⊗ ε(1 1 )1

  (1) (1 ) (2) (2 )

  = 1 ⊗ ε(1 )1

  (1) (2) (3)

  = 1 ⊗ 1

  (1) (2) H ) ∈ H⊗L H ) ∈ = ∆(1 H ).

  P♦rt❛♥t♦✱ t❡♠♦s ∆(1 ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♠♦str❛✲s❡ q✉❡ ∆(1 R ⊗ H

  ✳ ▲♦❣♦✱ t❡♠♦s ∆(1 H ) ∈ (H ⊗ L) ∩ (R ⊗ H) = R ⊗ L. ❆ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ s❡❣✉❡ ❞❡ ❬✶✶❪ ❧❡♠❛ ✶✳✹✳✺✳ ▼♦str❡♠♦s ❡♥tã♦✱ ❛ ❝♦♥❞✐✲ çã♦ (i) ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R ❡ l ∈ L✱ t❡♠♦s s R (ε R (t L (l))) = s R (ε R (1 ε(1 l))) ′ ′ (1) (2)

  = s R (1 ε(1 ε(1 l)1 )) ′ ′ (1 ) (1) (2) (2 ) = 1 ε(1 1 )ε(1 l)

  (1 ) (1) (2 ) (2) R

  = ⊓ (1 )ε(1 l)

  (1) (2)

  = 1 ε(1 l)

  (1) (2) ♣♦r (1)

  = t L (l), s L (ε L (t R (r))) = ε L (ε(r1 )1 ) ′ ′ (1) (2) = ε(1 ε(r1 )1 )1

  (1 ) (1) (2) (2 ) ′ ′

  = ε(r1 )ε(1 1 )1

  (1) (1 ) (2) (2 ) L

  = ε(r1 ) ⊓ (1 )

  (1) (2)

  = ε(r1 )1

  (1) (2) ♣♦r (1) R R L R R = t R (r), t (ε (s (l))) = t (ε (l))

  = t R (1 ε(l1 ))

  (1) (2) ′ ′

  = ε(1 ε(l1 )1 )1

  (1) (2) (1 ) (2 ) ′ ′

  = ε(l1 )ε(1 1 )1

  (2) (1) (1 ) (2 ) ′ ′

  = ε(l1 )1

  (1 ) (2 ) ♣♦r ✷✳✶✵ ♣á❣ ✻✷

  ′ ′

  = ε(ε(1 l)1 1 )1

  (1) (2) (1 ) (2 ) ′ ′

  = ε(ε(1 l)1 1 )1

  (1) (1 ) (2) (2 ) ♣♦r ✷✳✾ ♣á❣ ✻✶ ′ ′

  = ε(1 l)ε(1 1 )1

  (1) (1 ) (2) (2 ) L

  = ε(1 l) ⊓ (1 )

  (1) (2)

  = ε(1 l)1 ♣♦r (1) L (1) (2)

  = ⊓ (l) = l = s L (l),

  ❡ t❛♠❜é♠ ′ ′ t L (ε L (s R (r))) = t L (ε(1 r)1 )

  

(1 ) (2 )

′ ′

  = ε(1 ε(1 r)1 )1

  (2) (1 ) (2 ) (1)

′ ′

  = ε(1 1 )ε(1 r)1

  (2) (2 ) (1 ) (1)

  = ε(1 r)1

  (2) (1) ♣♦r ✷✳✶✵ ♣á❣ ✻✷

′ ′

  = ε(1 1 ε(r1 ))1

  (2) (1 ) (2 ) (1)

′ ′

  = ε(r1 )ε(1 1 )1

  (2 ) (2) (1 ) (1) ′ ′

  = ε(r1 )ε(1 1 )1

  (2 ) (1 ) (2) (1) ♣♦r ✷✳✾ ♣á❣ ✻✶

  = ε(r1 )1 R (2) (1) = ⊓ (r) = r = s R (r).

  ❆❣♦r❛ ♥♦t❡ q✉❡

  (∆⊗H)∆ π ⊗H H ⊗π L R

  // H ⊗ H ⊗ H // H ⊗ // H ⊗ H L H ⊗ H L H ⊗ R

  H, é ✐❣✉❛❧ ❛

  (H⊗∆)∆ H π H ⊗π ⊗ R L R

  H // H ⊗ H ⊗ H // H ⊗ H ⊗ R H // H ⊗ L H ⊗ R H. ❙❡❣✉❡ ❞❛í q✉❡

  (∆ L ⊗ R H)∆ R = (H ⊗ R π R )(π L ⊗ L H)(∆ ⊗ H)∆ = (π L ⊗ L H)(H ⊗ R π R )(H ⊗ ∆)∆ R ⊗ L H)∆ L = (H ⊗ R ∆ L )∆ R = (H ⊗ L ∆ R )∆ L .

  ◗✉❡ (∆ é ❛♥á❧♦❣♦✳ ▼♦str❡♠♦s ❛❣♦r❛ ❛ ❝♦♥❞✐çã♦ (iii) ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r h ∈ H

  ✱ r ∈ R ❡ l ∈ L✱ t❡♠♦s ′ ′ S(t L (l)ht R (r)) = S(1 ε(1 l)hε(r1 )1 )

  (1) (2) (1 ) (2 )

  ′ ′

  = ε(r1 )S(1 h1 )ε(1 l)

  (1 ) (1) (2 ) (2) ′ ′

  = ε(r1 )S(1 )S(h)S(1 )ε(1 l)

R L

(1 ) (2 ) (1) (2) = ⊓ (r)S(h) ⊓ (l)

  ♣♦r (2) ❡ (3) = s R (r)S(h)s L (l). ❆❣♦r❛ ♥♦t❡ q✉❡✱ ♣❛r❛ t♦❞♦ h ∈ H✱ t❡♠♦s

  µ L (S ⊗ L H)∆ L (h) = µ L (S(h ) ⊗ L h )

  (1) (2)

  = S(h )h R (1) (2) = ⊓ (h) = s R ◦ ε R (h), t❛♠❜é♠ t❡♠♦s

  µ R (H ⊗ R S)∆ R (h) = µ R (H ⊗ R S)(h ⊗ R h )

  (1) (2)

  = µ R (h ⊗ R S(h ))

  (1) (2)

  = h S(h )

  (1) (2)

L

  = ⊓ (h) = s L ◦ ε L (h). L , H R , S) P♦rt❛♥t♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ H = (H ✱ ❝♦♠♦ ❞❡✜♥✐❞♦s ♥❡ss❡ ❡①❡♠✲ ♣❧♦✱ é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳ e e ❊①❡♠♣❧♦ ✸✳✺ ✭❖ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ R ✮ ❏á ✈✐♠♦s q✉❡ R é ✉♠ R✲❜✐❛❧❣❡✲ e e

  , s R , t R ) R ❜ró✐❞❡ à ❞✐r❡✐t❛✱ ❝♦♠ ❡str✉t✉r❛ ❞❡ R ✲❛♥❡❧ (R ✱ ❡♠ q✉❡ t ❡ e s R

  , ∆ R , ε R ) sã♦ ❛s ✐♥❝❧✉sõ❡s ❡ ❝♦♠ ❡str✉t✉r❛ ❞❡ R✲❝♦❛♥❡❧ (R ✱ ❡♠ q✉❡ ∆ R (a ⊗ b) = (1 R ⊗ b) ⊗ R (a ⊗ 1 R ) R (a ⊗ b) = ba,

  ❡ ε ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ e op R ) cop op op ✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✽ ♣♦❞❡♠♦s ♠✉♥✐r (R ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ R

  ⊗ R op ✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛✳ ❆ ❡str✉t✉r❛ ❞❡ R ✲❛♥❡❧ é ❛ tr✐♣❧❛ op op op op (R ⊗ R, s R , t R ) R = s R R = t R op op ✱ ❡♠ q✉❡ s ❡ t ❡ ❛ ❡str✉t✉r❛ ❞❡ op op op op R ⊗ R, ∆ R , ε R ) R (a ⊗ R

  b) = ✲❝♦❛♥❡❧ é ❛ tr✐♣❧❛ (R ✱ ❡♠ q✉❡ ∆ op op

  (a ⊗ 1 R ) ⊗ R (1 R ⊗ b) R (a ⊗ b) = ba e op ❡ ε ✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ R✳ ≃ R ⊗ R

  ❆❣♦r❛✱ ✉t✐❧✐③❛♥❞♦ q✉❡ R ✱ t❡♠♦s q✉❡ ❡st❛s ❡str✉t✉r❛s✱ ❥✉♥t♦ e e −→ R

  ❝♦♠ ❛ ❛♥tí♣♦❞❛ S : R ✱ t❛❧ q✉❡ S(a ⊗ b) = b ⊗ a✱ ❝♦♥st✐t✉❡♠ ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳

  ❉❡ ❢❛t♦✱ ♥♦t❡ q✉❡ ❛s ❝♦♥❞✐çõ❡s ❞♦ ❛①✐♦♠❛ (i) ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ sã♦ tr✐✈✐❛❧♠❡♥t❡ s❛t✐s❢❡✐t❛s✳ P❛r❛ ♠♦str❛r ❛s ❞❡♠❛✐s ❝♦♥❞✐✲ e op

  −→ R ⊗ R a ⊗ b 7−→ b ⊗ a çõ❡s✱ ✈❛♠♦s ✉s❛r ❛ ✐❞❡♥t✐✜❝❛çã♦ R ✳ P♦rt❛♥t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ R✱ t❡♠♦s op op e e

  (∆ R ⊗ R R )∆ R (a ⊗ b) = (∆ R ⊗ R R )((1 R ⊗ b) ⊗ R (a ⊗ 1 R ))

  = ∆ R op (b ⊗ 1 R ) ⊗ R (a ⊗ 1 R ) = (b ⊗ 1 R ) ⊗ R op (1 R ⊗ 1 R ) ⊗ R (a ⊗ 1 R ),

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ (R e ⊗ R op ∆ R )∆ R op (b ⊗ a) = (R e ⊗ R op ∆ R )((b ⊗ a) ⊗ R op (1 R ⊗ a)

  = (b ⊗ a) ⊗ R op ∆ R (a ⊗ 1 R ) = (b ⊗ 1 R ) ⊗ R op (1 R ⊗ 1 R ) ⊗ R (a ⊗ 1 R ). ❆♥❛❧♦❣❛♠❡♥t❡✱ ♠♦str❛✲s❡ (∆ R ⊗ R op

  R e )∆ R op = (R e ⊗ R ∆ R op )∆ R ✳ ❆❣♦r❛✱

  ♣❛r❛ q✉❛✐sq✉❡r a, b, r ❡ s ∈ R✱ t❡♠♦s S(t R op (s) · op (a ⊗ b) · op t R (r)) = S((s ⊗ 1 R ) · op (a ⊗ b) · op (1 R ⊗ r))

  = S((1 R ⊗ r)(a ⊗ b)(s ⊗ 1 R )) = S(as ⊗ br) = br ⊗ as = br ⊗ s a = (1 R ⊗ s)(b ⊗ a)(r ⊗ 1 R ) = (r ⊗ 1 R ) · op (b ⊗ a) · op (1 R ⊗ s) = s R (r) · op S(a ⊗ b) · op s R op (s). ❚❛♠❜é♠ t❡♠♦s µ R op (S ⊗ R op R e )∆ R op (b ⊗ a) = µ R op (S ⊗ R op R e )((b ⊗ 1 R ) ⊗ R op (1 R ⊗ a))

  = µ R op (S(1 R ⊗ b) ⊗ R op (a ⊗ 1 R )) = (b ⊗ 1 R ) · op (a ⊗ 1 R ) = (a ⊗ 1 R )(b ⊗ 1 R ) = ab ⊗ 1 R ,

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ s R ◦ ε R (a ⊗ b) = s R (ba) = ba ⊗ 1 R . ❆♥❛❧♦❣❛♠❡♥t❡✱ ♠♦str❛✲s❡ µ R ◦ (R e

  ⊗ R S) ◦ ∆ R = s R op ◦ ε R op

  ❊①❡♠♣❧♦ ✸✳✻ ✭❖ t♦r♦ q✉â♥t✐❝♦ ❛❧❣é❜r✐❝♦✮ ❈♦♥s✐❞❡r❡ ✉♠❛ á❧❣❡❜r❛ T q s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k✱ ❣❡r❛❞❛ ♣♦r ❞♦✐s ❡❧❡♠❡♥t♦s ✐♥✈❡rsí✈❡✐s U ❡

  V ✱ s✉❥❡✐t♦s ❛ r❡❧❛çã♦ UV = qV U✱ ❡♠ q✉❡ q é ✉♠ ❡❧❡♠❡♥t♦ ✐♥✈❡rsí✈❡❧ ❡♠ k q ✳ T ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛ s♦❜r❡ ❛ s✉❜á❧❣❡❜r❛

  ❝♦♠✉t❛t✐✈❛ R✱ ❣❡r❛❞❛ ♣♦r U✳ ❖s ♠♦r✜s♠♦s s♦✉r❝❡ ❡ t❛r❣❡t sã♦ ❞❛❞♦s q R R ❛♠❜♦s ♣❡❧❛ ✐♥❝❧✉sã♦ R −→ T ✳ ❖ ❝♦♣r♦❞✉t♦ ∆ ❡ ❛ ❝♦✉♥✐❞❛❞❡ ε sã♦ ❞❛❞♦s ♣♦r m n m n m m n n

  ∆ R (V U ) = V U ⊗ R V ε R (V U ) = U , ❡♠ q✉❡ m, n ∈ Z✳ ❙✐♠❡tr✐❝❛♠❡♥t❡✱ ❡①✐st❡ ✉♠❛ ❡str✉t✉r❛ ❞❡ R✲❜✐❛❧❣❡❜ró✐❞❡ L L à ❡sq✉❡r❞❛ ❞❛❞❛ ♣❡❧♦ ❝♦♣r♦❞✉t♦ ∆ ❡ ❛ ❝♦✉♥✐❞❛❞❡ ε ✱ t❛✐s q✉❡ n m n m m n m n

  ∆ L (U V ) = U V ⊗ R V ε L (U n m −m n V ) = U .

  V ) = V U ❊st❛s ❡str✉t✉r❛s ❥✉♥t♦ ❝♦♠ ❛ ❛♥tí♣♦❞❛ S(U ✱ ❝♦♥st✐✲ t✉❡♠ ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳

  ❆ ❝♦♥❞✐çã♦ (i) ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ é tr✐✈✐❛❧♠❡♥t❡ s❛t✐s✲ m n U ∈ T q

  ❢❡✐t❛✳ ▼♦str❡♠♦s ❛ ❝♦♥❞✐çã♦ (ii)✳ ❉❡ ❢❛t♦✱ ♣❛r❛ V ✱ t❡♠♦s m n m n m (∆ L ⊗ R T q )∆ R (V U ) = (∆ L ⊗ R T q )(V U ⊗ R m n m V )

  = ∆ L (V U ) ⊗ R n m m

  V

  −mn

  = ∆ L (q U n m m m V ) ⊗ R

  V

  −mn

  = q U V ⊗ R V ⊗ R V , ♣♦r ♦✉tr♦ ❧❛❞♦✱ m n −mn n m

  (T q ⊗ R ∆ R )∆ L (V U ) = (T q ⊗ R ∆ R )∆ L (q U V )

  −mn n m m

  = (T q ⊗ R ∆ R )(q U V ⊗ R V )

  −mn n m m m n m = q U V ⊗ R V ⊗ R V .

  V ∈ T q ❚❛♠❜é♠✱ ♣❛r❛ t♦❞♦ U ✱ t❡♠♦s n m n m m

  (∆ R ⊗ R T q )∆ L (U V ) = (∆ R ⊗ R T q )(U n m m V ⊗ R V ) = ∆ R (U mn m n m m V ) ⊗ R

  V = q

  V U ⊗ R V ⊗ R V , ♣♦r ♦✉tr♦ ❧❛❞♦✱ n m mn m n m

  (T q ⊗ R ∆ L )∆ R (U V ) = (T q ⊗ R ∆ L )(q mn m n m m

  V U ⊗ R V ) l n m p k = q

  V U ⊗ R V ⊗ R V . , U V ∈ T q , U ∈ R

  ❆❣♦r❛✱ ♣❛r❛ q✉❛✐sq✉❡r V ❡ U ✱ t❡♠♦s n m p l −mp n p m l S(U

  V U V ) = S(q U U

  V V )

  −mp n +p m +l

  = S(q U V ) n

  −mp −(m+l) +p

  = q

  V U , ♣♦r ♦✉tr♦ ❧❛❞♦✱ p l n m p n

  −l −m

  S(U V )S(U V ) = V U p p n

  V U

  (−m) −l −m

  = q

  V V U U p

  −pm −(l+m) +n

  = q V U . ❉❡ss❛ ❢♦r♠❛✱ p n m l l n m p

  S(U U

  V U ) = S(U )S(U l n m p V )S(U ) = U S(U V )U . ▼♦str❡♠♦s q✉❡ ✈❛❧❡ ❛ ❝♦♥❞✐çã♦ (iv) ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ n m

  V ∈ T q t♦❞♦ U ✱ t❡♠♦s n m m n m

  

−m

  S(U V )V = V U

  V n m

  −(−m)n −m

  = q U mn n

  V V = q U

  ❡ t❛♠❜é♠ m n m m n −m

  V U S(V ) = V U m −m n −mn

  V = V

  V U q

  −mn n q = q U .

  ❙❡❣✉❡ ♣♦rt❛♥t♦✱ q✉❡ T é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳ ❊①❡♠♣❧♦ ✸✳✼ ✭❊①t❡♥sã♦ ❡s❝❛❧❛r✮ ❈♦♥s✐❞❡r❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢ H ❡ ✉♠❛ á❧❣❡❜r❛ A ❝♦♠✉t❛t✐✈❛ tr❛♥ç❛❞❛ ♥❛ ❝❛t❡❣♦r✐❛ ❞♦s ♠ó❞✉❧♦s ❞❡ ❨❡tt❡r✲ ❉r✐♥❢❡❧✬❞ ❞❡ H✱ ♦✉ s❡❥❛✱ A é ✉♠ H✲♠ó❞✉❧♦ á❧❣❡❜r❛ ❡ ✉♠ H✲❝♦♠ó❞✉❧♦ á❧❣❡❜r❛ q✉❡ s❛t✐s❢❛③✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ A✱

  (0) (1) b (a ⊳ b ) = ab.

  ❙❛❜❡♠♦s ❞❛ ❙❡çã♦ ✷✳✸✳✺ q✉❡ ♦ ♣r♦❞✉t♦ s♠❛s❤ A#H ♣♦ss✉✐ ✉♠❛ ❡str✉✲ e A A , t ) t✉r❛ ❞❡ A✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✱ ❝♦♠ ❡str✉t✉r❛ ❞❡ A ✲❛♥❡❧ (A#H, s ✱ A (a) = a #a A (a) = a#1 H (0) (1)

  ❡♠ q✉❡ s ❡ t ✱ ❡ ❝♦♠ ❡str✉t✉r❛ ❞❡ A✲ , ε )

  ❝♦❛♥❡❧ (A#H, ∆ A A ✱ ❡♠ q✉❡ ∆ (a#h) = (a#h ) ⊗ A (1 A #h ) (a#h) = aε(h), A (1) (2) ❡ ε A

  ♣❛r❛ q✉❛✐sq✉❡r a ∈ A ❡ h ∈ H✳ ❚❛♠❜é♠✱ ♣♦❞❡♠♦s ♠✉♥✐r A#H op ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ A ✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛✱ ❝♦♠ ❡str✉t✉r❛ ❞❡ op

op op op

(0) (1)

  A ⊗ A A , t A ) A (a) = a ⊳ S(a )#1 H ✲❛♥❡❧ (A#H, s ✱ ❡♠ q✉❡ s ✱ op op (0) (1) op op t A (a) = a #a

  , ε ) ✱ ❡ ❝♦♠ ❡str✉t✉r❛ ❞❡ A ✲❝♦❛♥❡❧ (A#H, ∆ A A ✱

  ❡♠ q✉❡ op op ∆ (a#h) = (a#h ) ⊗ A (1 A #h ) A (1) (2)

  ❡

  (0) −1 −1 (1)

  ε op (a#h) = a ⊳ S (hS (a )), A ♣❛r❛ q✉❛✐sq✉❡r a ∈ A ❡ h ∈ H✳ ❚❛♠❜é♠✱ t❡♠♦s ❛ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❡♥tr❡ ❛ ❛çã♦ ❡ ❛ ❝♦❛çã♦ ❞❡ H ❡♠ A✱ q✉❡ é ❞❛❞❛ ♣♦r

  

(0) (1) (o) (1)

(a ⊳ h) ⊗ (a ⊳ h) = a ⊳ h ⊗ S(h )a h .

  (2) (2) (3)

  ❊st❛s ❡str✉t✉r❛s ❥✉♥t♦ ❝♦♠ ❛ ❛♥tí♣♦❞❛

  (0) (1)

  S : A#H −→ A#H, a#h 7−→ a ⊳ S(h )#a S(h ),

  (2) (1)

  ❝♦♥st✐t✉❡♠ ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳ ▼♦str❡♠♦s ♣r✐♠❡✐r♦ q✉❡ ✈❛❧❡ ❛ ❝♦♥❞✐çã♦ (i) ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ a ∈ A✱ t❡♠♦s op op op op (0) (1) t A (ε A (s A (a))) = t A (ε A (a #a )) op (0)(0) −1 (1) −1 (0)(1)

  = t A (a ⊳ S (a S (a ))) op (0) −1 (1) −1 (1) = t A (a ⊳ S (a S (a )))

  (2) (1) op (0) −1 (1)

  = t A (a ⊳ S (ε(a )1 H )) op (0) (1) −1 = t A (a ε(a ) ⊳ S (1 H )) op = t A (a) = s A (a), op (0) (1) t A (ε A (s A (a))) = t A (ε A (a ⊳ S(a )#1 H ))

  (0) (1)

  = t A (a ⊳ S(a ))

  (0) (1)

  = a ⊳ S(a )#1 H op = s A (a), op (0) (1) s A (ε A (t A (a))) = s A (ε A (a #a ))

  (0) (1)

  = s A (a ε(a )) op = s A (a) = t A (a),

  ❡ t❛♠❜é♠ op op s A (ε A (t A (a))) op op = s A (ε A (a#1 H )) op (0) −1 −1 (1)

  = s A (a ⊳ S (1 H S (a )))

  (0) −2 (1) (0) (0) −2 (1) (1)

  = (a ⊳ S (a )) ⊳ S((a ⊳ S (a )) )#1 H

  (0)(0) −2 (1) −2 (1) (0)(1) −2 (1)

  = (a ⊳ S (a )) ⊳ S(S(S (a ))a S (a ))#1 H

  (2) (1) (3) (0)(0) −2 (1) −1 (1) (0)(1) −2 (1)

  = (a ⊳ S (a )) ⊳ S(S (a )a S (a ))#1 H

  (2) (1) (3) (0)(0) −2 (1) −1 (1) (0)(1) (1)

  = (a ⊳ S (a )) ⊳ S (a )S(a )a #1 H

  (2) (3) (1) (0)(0) −2 (1) −1 (1) (0)(1) (1)

  = a ⊳ (S (a )S (a )S(a )a )#1 H

  (2) (3) (1) (0)(0) −1 (1) −1 (1) (0)(1) (1)

  = a ⊳ (S (a S (a ))S(a )a )#1 H

  (3) (2) (1) (0) −1 (4) −1 (3) (1) (2)

  = a ⊳ (S (a S (a ))S(a )a )#1 H

  (0) −1 (3) (1) (2)

  = a ⊳ (S (ε(a )1 H )S(a )a )#1 H

  (0) −1 (1) (2) (3)

  = a ⊳ (S (1 H )S(a )a ε(a ))#1 H

  (0) (1) (2)

  = a ⊳ (S(a )a )#1 H

  (0) (1)

  = a ⊳ (ε(a )1 H )#1 H = a#1 H = t A (a).

  ❉❡♥♦t❡ H = A#H✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❛r❛ q✉❛✐sq✉❡r a ∈ A ❡ h ∈ H✱ t❡♠♦s op op H H (∆ A ⊗ A )∆ A (a#h) = (∆ A ⊗ A )((a#h ) ⊗ A (1 A #h )) op (1) (2)

  = ∆ A (a#h ) ⊗ A (1 A #h )

  (1) (2) op

  = (a#h ) ⊗ A (1 A #h ) ⊗ A (1 A #h ),

  (1) (2) (3)

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ op op op op (H ⊗ A ∆ A )∆ A (a#h) = (H ⊗ A ∆ A )((a#h ) ⊗ A (1 A #h )) op (1) (2) = (a#h ) ⊗ A (1 A #h ) ⊗ A (1 A #h ).

  (1) (2) (3) A ⊗ A )∆ A = (H ⊗ A ∆ A )∆ A op op op H

  ❆♥❛❧♦❣❛♠❡♥t❡✱ ♠♦str❛✲s❡ (∆ ✳ P❛r❛ ♠♦str❛r♠♦s q✉❡ ✈❛❧❡ ❛ ❝♦♥❞✐çã♦ (iii)✱ ♠♦str❡♠♦s ❛♥t❡s q✉❡ S é ❛♥t✐✲ ♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ❉❡ ❢❛t♦✱ q✉❛✐sq✉❡r a, b ∈ A ❡ h, k ∈ H✱ t❡♠♦s S((a#h)(b#k)) = S(b(a ⊳ k )#hk )

  (1) (2) (0) (1)

  = (b(a ⊳ k )) ⊳ S(h k )#(b(a ⊳ k )) S(h k )

  (1) (2) (3) (1) (1) (2)

  (0) (0) (1) (1)

  = (b (a ⊳ k ) ) ⊳ S(h k )#b (a ⊳ k ) S(h k )

  

(1) (2) (3) (1) (1) (2)

(0) (0) (1) (1)

  = (b (a ⊳ k )) ⊳ S(h k )#b S(k )a k S(h k )

  (1)(2) (2) (3) (1)(1) (1)(3) (1) (2) (0) (0) (1) (1)

  = (b (a ⊳ k )) ⊳ S(h k )#b S(k )a k S(h k )

  (2) (2) (5) (1) (3) (1) (4) (0) (0) (1) (1)

  = (b (a ⊳ k )) ⊳ S(h k )#b S(k )a k S(k )S(h )

  (2) (2) (5) (1) (3) (4) (1) (0) (0) (1) (1)

  = (b (a ⊳ k )) ⊳ S(h k )#b S(k )a ε(k )1 H S(h )

  (2) (2) (4) (1) (3) (1) (0) (0) (1) (1)

  = (b (a ⊳ k )) ⊳ S(h ε(k )k )#b S(k )a

  1 H S(h )

  (2) (2) (3) (4) (1) (1) (0) (0) (1) (1)

  = (b (a ⊳ k )) ⊳ S(h k )#b S(k )a S(h )

  

(2) (2) (3) (1) (1)

(0) (0) (1) (1)

  = (b ⊳ S(h k ))((a ⊳ k ) ⊳ S(h k ))#b S(k )a S(h )

  (3) (4) (2) (2) (3) (1) (1) (0) (0) (1) (1)

  = (b ⊳ S(h k ))(a ⊳ k S(k )S(h ))#b S(k )a S(h )

  (3) (4) (2) (3) (2) (1) (1) (0) (0) (1) (1)

  = (b ⊳ S(h k ))(a ⊳ ε(k )1 H S(h ))#b S(k )a S(h )

  (3) (3) (2) (2) (1) (1) (0) (0) (1) (1)

  = (b ⊳ S(h ε(k )k ))(a ⊳ S(h ))#b S(k )a S(h )

  (3) (2) (3) (2) (1) (1) (0) (0) (1) (1)

  = (b ⊳ S(h k ))(a ⊳ S(h ))#b S(k )a S(h ),

  

(3) (2) (2) (1) (1)

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ S(b#k)S(a#h)

  (0) (1) (0) (1)

  =(b ⊳ S(k )#b S(k ))(a ⊳ S(h )#a S(h ))

  (2) (1) (2) (1) (0) (0) (1) (1) (1)

  =(a ⊳S(h ))((b ⊳S(k ))⊳ a S(h ))#b S(k )a S(h )

  (3) (2) (1) (2) (1) (2) (1) (0)(0) (0) (0)(1) (1) (1)

  =(a ⊳S(h ))((b ⊳S(k ))⊳ a S(h ))#b S(k )a S(h )

  (3) (2) (2) (1) (1) (0)(0) (0) (0)(1) (1) (1)

  =(a ⊳S(h ))(((b ⊳S(k ))⊳a )⊳S(h ))#b S(k )a S(h )

  (3) (2) (2) (1) (1) (0)(0) (0) (0)(1) (1) (1)

  = ((a ((b ⊳ S(k )) ⊳ a )) ⊳ S(h ))#b S(k )a S(h )

  (2) (2) (1) (1) (0) (0) (1) (1)

  = ((b ⊳ S(k ))a ⊳ S(h ))#b S(k )a S(h )

  (2) (2) (1) (1) (0) (0) (1) (1)

  = (b ⊳ S(k )S(h ))(a ⊳ S(h ))#b S(k )a S(h )

  (2) (3) (2) (1) (1) (0) (0) (1) (1)

  = (b ⊳ S(h k ))(a ⊳ S(h ))#b S(k )a S(h ).

  

(3) (2) (2) (1) (1)

  ❙❡❣✉❡ q✉❡ S é ❛♥t✐✲♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ❆❣♦r❛ ♥♦t❡ q✉❡✱ ♣❛r❛ t♦❞♦ a ∈ A✱ t❡♠♦s

  (0) (1)

  S(t A (a)) = S(a#1 H ) = a #a = s A (a), t❛♠❜é♠ t❡♠♦s op (0) (1) S(t A (a)) = S(a #a )

  (0)(0) (1) (0)(1) (1)

  = a ⊳ S(a )#a S(a )

  (2) (1)

  (0) (1) (1) (1)

  = a ⊳ S(a )#a S(a )

  (3) (1) (2)

(0) (1) (1)

  = a ⊳ S(a )#ε(a )1 H

  (2) (1)

(0) (1) (1)

  = a ⊳ S(ε(a )a )#1 H

  (1) (2) (0) (1)

  = a ⊳ S(a )#1 H op = s A (a). ❙❡❣✉❡ ♣♦rt❛♥t♦✱ ♣❛r❛ q✉❛✐sq✉❡r a, b, c ∈ A ❡ h ∈ H✱ q✉❡ op op

  S(t A (a)(c#h)t A (b)) = S(t A (b))S(c#h)S(t A (a)) op = s A (b)S(c#h)s A (a). P❛r❛ ✜♥❛❧✐③❛r✱ ♠♦str❡♠♦s q✉❡ ✈❛❧❡ ❛ ❝♦♥❞✐çã♦ (iv)✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐s✲ q✉❡r a ∈ A ❡ h ∈ H✱ t❡♠♦s

  µ A (H ⊗ A S)∆ A (a#h) = µ A (H ⊗ A S)((a#h ) ⊗ A (1 A #h ))

  (1) (2)

  = (a#h )S(1 A #h )

  (1) (2)

  = (a#h )(1 A ⊳ S(h )#S(h ))

  (1) (3) (2)

  = (a#h )(ε(S(h ))1 A #S(h ))

  (1) (3) (2)

  = (a#h )(ε(h )1 A #S(h ))

  (1) (3) (2)

  = (a#h )(1 A #S(h ε(h )))

  (1) (2) (3)

  = (a#h )(1 A #S(h ))

  (1) (2)

  = a ⊳ S(h )#h S(h )

  (3) (1) (2)

  = a ⊳ S(h )#ε(h )1 H

  (2) (1)

  = a ⊳ S(ε(h )h )#1 H

  (1) (2)

  = a ⊳ S(h)#1 H , ♣♦r ♦✉tr♦ ❧❛❞♦✱ op op s A ◦ ε A (a#h) op (0) −1 −1 (1)

  = s A (a ⊳ S (hS (a )))

  (0) −1 −1 (1) (0) (0) −1 −1 (1) (1)

  = (a ⊳ S (hS (a ))) ⊳ S((a ⊳ S (hS (a ))) )#1 H

  

(0)(0) −1 −1 (1) −1 −1 (1) (0)(1)

  = (a ⊳ S (h S (a ))S(S(S (h S (a )))a

  (2) (2) (3) (1) −1 −1 (1)

  S (h S (a ))))#1 H

  (1) (3) (0) −2 (3) −1 −1 (2) (1) −1 (4)

  = (a ⊳ S (a )S (h )S(h S (a )a S(h S (a ))))#1 H

  (2) (3) (1) (0) −2 (3) −1 −1 (4) (1) (2)

  = (a ⊳ S (a )S (h )h S (a )S(a )a S(h ))#1 H

  (2) (1) (3) (0) −2 (2) −1 (3) (1)

  = (a ⊳ S (a )ε(h )1 H S (a )ε(a )1 H S(h ))#1 H

  (1) (2)

  (0) −2 (1) (2) −1 (3)

  = (a ⊳ S (ε(a )a )S (a )S(ε(h )h ))#1 H

  (1) (2) (0) −2 (1) −1 (2)

  = (a ⊳ S (a )S (a )S(h))#1 H

  (0) −1 (2) −1 (1)

  = (a ⊳ S (a S (a ))S(h))#1 H

  (0) −1 (1)

  = (a ⊳ S (ε(a )1 H )S(h))#1 H

  (0) (1) −1

  = (a ε(a ) ⊳ S (1 H )S(h))#1 H = (a ⊳ S(h))#1 H . ❚❛♠❜é♠ t❡♠♦s op op op op op op H H

  µ R (S ⊗ A )∆ R (a#h) = µ R (S ⊗ A )((a#h )⊗ A (1 A #h ))

  (1) (2)

  = S(a#h )(1 A #h )

  (1) (2) (0) (1)

  = (a ⊳ S(h ))#a S(h ))(1 A #h )

  (2) (1) (3) (0) (1)

  = (a ⊳ S(h )h )#a S(h )h

  (2) (3) (1) (4) (0) (1)

  = (a ⊳ ε(h )1 H )#a S(h )h

  (2) (1) (3) (0) (1)

  = (a ⊳ 1 H )#a S(h )ε(h )h

  (1) (2) (3)

(0) (1)

  = a #a S(h )h

  (1) (2)

(0) (1)

  = a #a ε(h)1 H = s R (aε(h)) = s R (ε R (a#h)).

  

✸✳✷ Pr♦♣r✐❡❞❛❞❡s ❇ás✐❝❛s ❞❡ ❍♦♣❢ ❆❧❣❡❜ró✐✲

❞❡s

  ◆❡st❛ s❡çã♦ ✈❡r❡♠♦s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ✐♥❞✐❝❛♠ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❡s♣❡r❛❞♦ ❞❛ ❛♥tí♣♦❞❛ ❞❡ ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✱ ❝♦♠ r❡s♣❡✐t♦ ❛s ❡str✉t✉r❛s L , H R , S) ❞❡ ❛♥❡❧ ❡ ❝♦❛♥❡❧ s✉❜❥❛❝❡♥t❡s✳ ❈♦♥s✐❞❡r❡ H = (H ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ s♦❜r❡ á❧❣❡❜r❛s ❞❡ ❜❛s❡s L ❡ R✳ ❙❡❣✉❡ ❞♦ ❛①✐♦♠❛ (i) ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ q✉❡ L ❡ R sã♦ ❛♥t✐✲✐s♦♠♦r❢❛s✳ ❉❡ ❢❛t♦✱ ❡①✐st❡♠ ✐s♦♠♦r✜s♠♦s op op ε L ◦ s R : R −→ L R ◦ t L : L −→ R .

  ❡ ε ✭✸✳✶✮ ❙✐♠❡tr✐❝❛♠❡♥t❡✱ ❡①✐st❡♠ ✐s♦♠♦r✜s♠♦s ✐♥✈❡rs♦s op op ε R ◦ s L : L −→ R L ◦ t R : R −→ L .

  ❡ ε ✭✸✳✷✮

  ′

  ∈ R ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r, r ✱ t❡♠♦s

  ′ ′

  ε L ◦ s R (rr ) = ε L (s R (rr ))

  = ε L (t L ◦ ε L ◦ s R (r)s R (r

  t L (l))s L (ε L (h

  (2) (1)

  t L (l))h

  (1)

  )) = S(h

  (2)

  t L (l))s L (ε L (h

  (1)

  )) = S(h

  (2)

  (1)

  (2) (2)

  t L (l)) = S(h

  (1)

  ))h

  

(2)

  S(ht L (l)) = S(t L (ε L (h

  )s R (r) = S(h)s R (r), t❛♠❜é♠ t❡♠♦s

  (1)

  ))h

  

(2)

  ))s R (r) = S(t L (ε L (h

  S(h

  ) = S(h

  )s L (ε L (h

  

(1)

  (2)

  ))S(h

  (1)

  ) = t R (ε R (s L (l)))s R (ε R (h

  (2)

  ))t R (ε R (s L (l)))S(h

  

(1)

  ) = s R (ε R (h

  (2)

  )s L (l)S(h

  ) = s R ◦ ε R (h

  (1) (1)

  (2)

  s L (l)S(h

  

(1)

(2)

  )h

  (1) (1)

  ) = S(h

  (2)

  S(h

  (1) (2)

  t L (l))h

  (2)

  (1)

  ′

  

(2)

  

(1)

(2)

  )h

  (1) (1)

  ) = S(h

  (2)

  ))S(t R (r)h

  

(1)

  ))) = s R (ε R (h

  (1)

  t R (ε R (h

  ♣❛r❛ q✉❛✐sq✉❡r r ∈ R✱ l ∈ L ❡ h ∈ H✳ ❉❡ ❢❛t♦✱ S(t R (r)h) = S(t R (r)h

  (2)

  S(t R (r)ht L (l)) = s L (l)S(h)s R (r), ✭✸✳✹✮

  ✭✸✳✸✮ ❆❣♦r❛✱ s❡❣✉❡ ❞♦s ❛①✐♦♠❛s (ii)✱ (iii) ❡ (iv) ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❍♦♣❢ ❛❧❣❡✲ ❜ró✐❞❡✱ q✉❡

  ′ )ε L ◦ s R (r).

  ))(ε L ◦ s R (r)) = ε L ◦ s R (r

  ′

  ) · (ε L ◦ s R (r))) = ε L (s R (r

  ′

  )) = ε L (s R (r

  ′

  )) = ε L (t L (ε L ◦ s R (r))s R (r

  S(t R (r)h

  ) = S(h

  ))t L (ε L (s R (r))) = S(h

  (1)

  (2)

  )s L (ε L (h

  (1)

  )) = S(h

  (2)

  )t L (ε L (s R (r)))s L (ε L (h

  (1)

  )) = S(h

  (2)

  )s R (r)s L (ε L (h

  ) = S(h

  (1)

  (2) (2)

  S(h

  (2) (1)

  )s R (r)h

  (1)

  ) = S(h

  (2) (2)

  S(t R (r)h

  

(2)

(1)

  )h

  )

  (1) (2)

  = s L (l)s R (ε R (h ))S(h )

  (2) (1)

  = s L (l)S(h t R (ε R (h ))) = s L (l)S(h). Pr♦♣♦s✐çã♦ ✸✳✽ ❙❡❥❛ H ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ s♦❜r❡ á❧❣❡❜r❛s ❜❛s❡s L ❡ R e ✱ ❡ H ❛ k✲á❧❣❡❜r❛ s✉❜❥❛❝❡♥t❡✳ ❊♥tã♦✱ ❛ ❛♥tí♣♦❞❛ S é ✉♠ ♠♦r✜s♠♦ ❞❡ R

  ✲❛♥é✐s op (H, s R , t R ) −→ (H , s L ◦ (ε L ◦ s R ), t L ◦ (ε L ◦ s R )). e ❆♥❛❧♦❣❛♠❡♥t❡✱ t❛♠❜é♠ é ✉♠ ♠♦r✜s♠♦ ❞❡ L ✲❛♥é✐s op (H, s L , t L ) −→ (H , s R ◦ (ε R ◦ s L ), t R ◦ (ε R ◦ s L )). ❊♠ ♣❛rt✐❝✉❧❛r✱ S : H −→ H é ❛♥t✐♠♦r✜s♠♦ ❞❡ k✲á❧❣❡❜r❛s✳ op

  , s L ◦(ε L ◦s R ), t L ◦(ε L ◦ ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r♦✱ ✈❛♠♦s ✈❡r q✉❡ (H e s R )) L ◦ s R : R −→ L

  é ✉♠ R ✲❛♥❡❧✳ ❉❡ ❢❛t♦✱ ♣♦r ✸✳✸ ε é ❛♥t✐♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ✱ s❡❣✉❡ q✉❡

  ′ ′

  (s L ◦ (ε L ◦ s R ))(rr ) = s L ((ε L ◦ s R )(rr ))

  ′

  = s L ((ε L ◦ s R )(r )(ε L ◦ s R )(r))

  ′

  = s L ◦ (ε L ◦ s R )(r )(s L ◦ (ε L ◦ s R )(r))

  ′

  = s L ◦ (ε L ◦ s R )(r) · op (s L ◦ (ε L ◦ s R )(r )), t❛♠❜é♠ t❡♠♦s

  ′ ′

  (t L ◦ (ε L ◦ s R ))(rr ) = t L ((ε L ◦ s R )(rr ))

  ′

  = t L ((ε L ◦ s R )(r )(ε L ◦ s R )(r))

  ′

  = t L ((ε L ◦ s R )(r))t L ((ε L ◦ s R )(r ))

  ′ L ◦ (ε L ◦ s R ) L ◦ (ε L ◦ s R ) = t L ((ε L ◦ s R )(r )) · op t L ((ε L ◦ s R )(r)).

  ➱ ❝❧❛r♦ q✉❡ s ❡ t ❝♦♠✉t❛♠ ♥❛s ✐♠❛❣❡♥s✳ ❙❡❣✉❡ op e ♣♦rt❛♥t♦✱ q✉❡ H é ✉♠ R ✲❛♥❡❧✳ ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ S é ♠♦r✜s♠♦ e

  ′

  ∈ R ❞❡ R ✲❜✐♠ó❞✉❧♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r, r ❡ h ∈ H✱ t❡♠♦s

  ′ ′ R R

  S((r ⊗ r ) · h) = S(s (r)t (r )h)

  ′

  = S(t L ◦ ε L ◦ s R (r)t R (r )h)

  ′

  = S(h)s L ◦ (ε L ◦ s R )(r)s R (r )

  ′

  = S(h)s L ((ε L ◦ s R )(r))s R (r )

  ′

  = S(h)s R (r )s L ((ε L ◦ s R )(r))

  ′

  = S(h)t L (ε L ◦ s R (r ))s L ((ε L ◦ s R )(r))

  ′

  = s L ◦ (ε L ◦ s R )(r) · op t L ◦ (ε L ◦ s R )(r ) · op S(h)

  ′

  = (r ⊗ r ) · S(h), t❛♠❜é♠ t❡♠♦s

  

  S(h · (r ⊗ r )) = S(hs R (r)t R (r ))

  ′

  = S(ht L ◦ (ε L ◦ s R (r))t R (r ))

  ′

  = s L ◦ (ε L ◦ s R (r))s R (r )S(h)

  ′

  = s L ◦ (ε L ◦ s R (r))t L ◦ (ε L ◦ s R (r ))S(h)

  

  = t L ◦ (ε L ◦ s R (r ))s L ◦ (ε L ◦ s R (r))S(h)

  ′

  = S(h) · op s L ◦ (ε L ◦ s R (r)) · op t L ◦ (ε L ◦ s R (r ))

  ′ = S(h) · (r ⊗ r ).

  ▼♦str❡♠♦s q✉❡ S é ❛♥t✐♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r h, k ∈ H

  ✱ t❡♠♦s S(hk) = S(t L (ε L (h ))h k)

  (2) (1)

  = S(h k)s L (ε L (h ))

  (1) (2)

  = S(h t L (ε L (k ))k )s L (ε L (h ))

  (1) (2) (1) (2) (1) (2)

  = S(h t L (ε L (k ))k )h S(h )

  (1) (2) (1) (2) (2) (1) (1) (2)

  = S(h t L (ε L (k ))k )h S(h )

  (1) (2) (1) (2) (1) (1) (2)

  = S(h k )h s L (ε L (k ))S(h )

  

(1) (1) (2) (2)

(1) (1) (1) (2) (2)

  = S(h k )h k S(k )S(h )

  

(1) (1) (2) (2) (2)

(1) (1) (1) (1) (2) (2)

  = S(h k )h k S(k )S(h )

  

(1) (1) (2) (2)

(1) (1) (1) (1) (2) (2)

  = S((h k ) )(h k ) S(k )S(h )

  (1) (2) (1) (1) (2) (2)

  = s R ◦ ε R (h k )S(k )S(h )

  

(2) (1) (1) (2)

  = S(k t R ◦ ε R (h k ))S(h )

  (2) (1) (1) (2)

  = S(k t R (ε R (s R (ε R (h ))k )))S(h )

  (1) (2) (1) (2)

  = S(t R (ε R (h ))k t R (ε R (k )))S(h )

  (1) (2)

  = S(k)s R (ε R (h ))S(h )

  (2) (1)

  = S(k)S(h t R (ε R (h ))) = S(k)S(h). H = s R ◦ ε R (1 H ) = S(1 H )1 H = S(1 H ) ❆❣♦r❛ ♥♦t❡ q✉❡ 1 ✳ ❉❡✜♥❛ e ′ ′

  η : R −→ H, η(r ⊗ r ) = s R (r)t R (r )

  ❡ e op \ ′

  ′ η : R −→ H , η(r ⊗ r ) = s L ◦ (ε L ◦ s R )(r) · op t L ◦ (ε L ◦ s R )(r ).

  b

  ′

  ∈ R ▼♦str❡♠♦s ❛ss✐♠✱ q✉❡ S ◦ η = bη✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r, r ✱ t❡♠♦s

  ′ ′

  S(η(r ⊗ r )) = S(s R (r)t R (r ))

  ′

  = S(t L ◦ (ε L ◦ s R )(r)t R (r ))

  ′

  = s R (r )S(1 H )s L ◦ (ε L ◦ s R )(r)

  ′

  = t L ◦ (ε L ◦ s R )(r )s L ◦ (ε L ◦ s R )(r)

  ′

  = s L ◦ (ε L ◦ s R )(r) · op t L ◦ (ε L ◦ s R )(r )

  ′ = η(r ⊗ r ).

  b e ❙❡❣✉❡ ♣♦rt❛♥t♦✱ q✉❡ S é ♠♦r✜s♠♦ ❞❡ R ✲❛♥é✐s✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♠♦str❛✲ e s❡ q✉❡ S é ♠♦r✜s♠♦ ❞❡ L ✲❛♥é✐s✳

  ❙❛❜❡♠♦s q✉❡ ♦ ♦♣♦st♦✲❝♦✲♦♣♦st♦ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢ H é t❛♠✲ ❜é♠ ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢✱ ❝♦♠ ❛ ♠❡s♠❛ ❛♥tí♣♦❞❛ S✳ ❙❡ S é ❜✐❥❡t✐✈❛✱ ❡♥tã♦ ♦ ♦♣♦st♦ ❡ ♦ ❝♦✲♦♣♦st♦ sã♦ á❧❣❡❜r❛s ❞❡ ❍♦♣❢✱ ✈❡r ✭❬✶✶❪✱ ❘❡♠❛r❦ ✹✳✷✳✶✵✮✳ ❙❡❣✉❡ ✉♠ ❛♥á❧♦❣♦ ❞❡st❡s ❢❛t♦s ♣❛r❛ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s✳ L , H R , S) Pr♦♣♦s✐çã♦ ✸✳✾ ❙❡❥❛ H = (H ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ s♦❜r❡ á❧✲ ❣❡❜r❛s ❜❛s❡s L ❡ R✱ ❡ H ❛ k✲á❧❣❡❜r❛ s✉❜❥❛❝❡♥t❡✳ ❊♥tã♦✱ ✈❛❧❡♠ ❛s s❡✲ ❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿ op op

  (1) R ) , (H L ) , S) ❆ tr✐♣❧❛ ((H cop cop é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ s♦❜r❡ á❧❣❡✲ op op ❜r❛s ❜❛s❡s R ❡ L ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡❀ op op −1

  (2) R ) , (H L ) , S ) ❙❡ ❛ ❛♥tí♣♦❞❛ S é ❜✐❥❡t✐✈❛✱ ❡♥tã♦ ((H é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ s♦❜r❡ á❧❣❡❜r❛s ❜❛s❡s R ❡ L✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ L ) cop , (H R ) cop , S ) −1 t❛♠❜é♠✱ ((H é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ s♦❜r❡ á❧✲ op op ❣❡❜r❛s ❜❛s❡s L ❡ R ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

  ❉❡♠♦♥str❛çã♦✿ (1) ❙❛❜❡♠♦s ❞❛ ♣r♦♣♦s✐çã♦ ✷✳✽ ❡ ❞❡ s❡✉ ❛♥á❧♦❣♦ ♣❛r❛ R ) op op ❜✐❛❧❣❡❜ró✐❞❡s à ❡sq✉❡r❞❛✱ q✉❡ (H cop é ✉♠ R ✲❜✐❛❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛ L op op op

  ) cop ⊗ R

  ❡ (H é ✉♠ L ✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❆ ❡str✉t✉r❛ ❞❡ R ✲ R ) , s R , t R ) op op op ❛♥❡❧ ❡♠ (H cop ✱ é ❞❛❞❛ ♣♦r (H cop ❡ ❛ ❡str✉t✉r❛ ❞❡ R ✲❝♦❛♥❡❧ R ) cop , ∆ , ε R ) op cop ❡♠ (H cop é ❞❛❞❛ ♣♦r (H ✳ ❙✐♠❡tr✐❝❛♠❡♥t❡✱ ❛ ❡str✉t✉r❛ op op op R

  ⊗ L L ) , s L , t L ) ❞❡ L ✲❛♥❡❧ ❡♠ (H cop ✱ é ❞❛❞❛ ♣♦r (H cop ❡ ❛ ❡str✉t✉r❛ op op L ) cop , ∆ , ε L ) cop ❞❡ L ✲❝♦❛♥❡❧ ❡♠ (H cop é ❞❛❞❛ ♣♦r (H ✳ ▼♦str❡♠♦s ❡♥✲ L tã♦ q✉❡ ✈❛❧❡♠ ♦s ❛①✐♦♠❛s ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳ ❉❡ ❢❛t♦✱ ♦ ❛①✐♦♠❛ (i)

  é tr✐✈✐❛❧♠❡♥t❡ s❛t✐s❢❡✐t♦✳ P❛r❛ ♠♦str❛r ♦ ❛①✐♦♠❛ (ii)✱ ✈❛♠♦s ✉s❛r q✉❡ op op H ⊗ L H ⊗ R H R H ⊗ L H

  é ✐s♦♠♦r❢♦ ❛ H ⊗ ❝♦♠♦ k✲♠ó❞✉❧♦s✳ ❖s r❡s♣❡❝t✐✈♦s ❜❛❧❛♥❝❡❛♠❡♥t♦s ♥♦s ♣r♦❞✉t♦s t❡♥s♦r✐❛✐s ❛❝✐♠❛ ❢♦r❛♠ ♠♦s✲ tr❛❞♦s ♥❛ ♣r♦♣♦s✐çã♦ ✷✳✽✳ ❆ss✐♠✱ ♣❛r❛ t♦❞♦ h ∈ H✱ t❡♠♦s cop cop cop op op op

  (∆ ⊗ L H)∆ (h) = (∆ ⊗ L H)(h ⊗ L h ) R L R

cop

op (2) (1) = ∆ (h ) ⊗ L h

R

(2) (1) (2) (1) op op

  = h ⊗ R h ⊗ L h

  (2) (2) (1) (2) (1) (1) op op

  = h ⊗ R h ⊗ L h ,

  (2) (1)

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ cop cop cop

op op op

(2) (1) (H ⊗ R ∆ )∆ (h) = (H ⊗ R ∆ )(h ⊗ R h ) L R L cop

  (2) (1) op

  = h ⊗ R ∆ (h ) L

  (2) (1) (1) op op = h ⊗ R h ⊗ L h . cop cop cop cop op op (2) (1)

  ⊗ R H)∆ = (H ⊗ L ∆ )∆ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♠♦str❛✲s❡ (∆ ✳ L R R L ❆❣♦r❛✱ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R✱ l ∈ L ❡ h ∈ H✱ t❡♠♦s

  S(t R (r) · op h · op t L (l)) = S(t L (l)ht R (r)) = s R (r)S(h)s L (l) = s L (l) · op S(h) · op s R (r).

  ❚❛♠❜é♠ t❡♠♦s cop op op op op op (2) (1) µ R (H ⊗ R S)∆ (h) = µ R (H ⊗ R S)(h ⊗ R h ) R

  (2) (1)

  = h · op S(h )

  (1) (2)

  = S(h )h = s L ◦ ε L (h). L (H ⊗ L S)∆ = s R ◦ε R op op cop ❆♥❛❧♦❣❛♠❡♥t❡✱ ♠♦str❛✲s❡ q✉❡ µ L ✳ P♦rt❛♥t♦✱ R ) , (H L ) , S) op op ❝♦♥❝❧✉í♠♦s q✉❡ ((H cop cop é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳ op (2) R )

  ❙❛❜❡♠♦s ❞❛ Pr♦♣♦s✐çã♦ ✷✳✽ q✉❡ (H é ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❡s✲ e op op R R R ) , t , s ) q✉❡r❞❛✳ ❆ ❡str✉t✉r❛ ❞❡ R ✲❛♥❡❧ ❡♠ (H é ❞❛❞❛ ♣♦r (H ❡ R ) R , ε R ) op

  ❛ ❡str✉t✉r❛ ❞❡ R✲❝♦❛♥❡❧ ❡♠ (H é ❞❛❞❛ ♣♦r (H, ∆ ✳ ❆♥❛❧♦❣❛✲ L ) op e ♠❡♥t❡✱ (H é ✉♠ L✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✳ ❆ ❡str✉t✉r❛ ❞❡ L ✲❛♥❡❧ L ) , t L , s L ) L ) op op op ❡♠ (H é ❞❛❞❛ ♣♦r (H ❡ ❛ ❡str✉t✉r❛ ❞❡ L✲❝♦❛♥❡❧ ❡♠ (H L , ε L ) é ❞❛❞❛ ♣♦r (H, ∆ ✳ ▼♦str❡♠♦s q✉❡ ✈❛❧❡♠ ♦s ❛①✐♦♠❛s ❞❡ ❍♦♣❢ ❛❧✲ ❣❡❜ró✐❞❡✳ ❖s ❛①✐♦♠❛s (i) ❡ (ii) sã♦ tr✐✈✐❛❧♠❡♥t❡ s❛t✐s❢❡✐t♦s✳ ▼♦str❡♠♦s q✉❡ ✈❛❧❡ ♦ ❛①✐♦♠❛ (iii)✳ ❆♥t❡s✱ ♥♦t❡ q✉❡ s❡ S é ❛♥t✐♠✉❧t✐♣❧✐❝❛t✐✈♦✱ t❡✲

  −1

  ♠♦s q✉❡ S t❛♠❜é♠ é ❛♥t✐♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r h ❡ k ∈ H✱ t❡♠♦s

  −1 −1 −1 −1

  S (hk) = S (S(S (h))S(S (k)))

  −1 −1 −1

  = S (S(S (k)S (h)))

  −1 −1 = S (k)S (h).

  ❆❣♦r❛ ♥♦t❡ q✉❡ ❞♦ ❛①✐♦♠❛ (iii) ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ ❡ ❞❡ ✸✳✹✱ t❡♠♦s S(t R (r)) = s R (r) L (l)) = s L (l),

  ❡ S(t ♣❛r❛ q✉❛✐sq✉❡r l ∈ L ❡ r ∈ R✳ ❙❡❣✉❡ q✉❡

  −1 −1

  S (s R (r)) = S (S(t R (r))) = t R (r) ❡

  −1 −1 S (s L (l)) = S (S(t L (l))) = t L (l).

  P♦rt❛♥t♦✱ ♣❛r❛ q✉❛✐sq✉❡r l ∈ L✱ r ∈ R ❡ h ∈ H✱ t❡♠♦s

  −1 −1

  S (s R (r) · op h · op s L (l)) = S (s L (l)hs R (r))

  −1 −1 −1 R L

  = S (s (r))S (h)S (s (l))

  −1

  = t R (r)S (h)t L (l)

  −1

  = t L (l) · op S (h) · op t R (r), op op R ) , (H L ) , S ) −1 ♦✉ s❡❥❛✱ ✈❛❧❡ ♦ ❛①✐♦♠❛ (iii) ♣❛r❛ ((H ✳ ▼♦str❡♠♦s

  (1) (2)

  S(h ) = s L ◦ ε L (h) ❛❣♦r❛ q✉❡ ✈❛❧❡ ♦ ❛①✐♦♠❛ (iv)✳ ❉❡ ❢❛t♦✱ ❞❡ h ✱ t❡♠♦s

  −1 (1) (2) (2) −1 (1)

  S (h ) · op h = h S (h )

  −1 (1) (2)

  = S (h S(h ))

  −1

  = S (s L (ε L (h))) = t L (ε L (h)),

  ♣❛r❛ t♦❞♦ h ∈ H✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♠♦str❛✲s❡

  −1 h · op S (h ) = t R ◦ ε R . (1) (2) op op R ) , (H L ) , S ) −1

  ❙❡❣✉❡ ♣♦rt❛♥t♦✱ q✉❡ ((H é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳

  L , H R , S)

  Pr♦♣♦s✐çã♦ ✸✳✶✵ ❙❡❥❛ H = (H ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳ ❊♥✲ L ◦ s R ) tã♦✱ ♦ ♣❛r (S, ε é ✉♠ ♠♦r✜s♠♦ ❞❡ ❜✐❛❧❣❡❜ró✐❞❡s à ❡sq✉❡r❞❛ op (H R ) −→ H L . cop L ◦s R ❉❡♠♦♥str❛çã♦✿ ❏á s❛❜❡♠♦s q✉❡ ε ❡ S sã♦ ♠♦r✜s♠♦s ❞❡ á❧❣❡❜r❛s✳ ❆❣♦r❛ t❡♠♦s q✉❡ s L ◦ (ε L ◦ s R ) = S ◦ t L ◦ (ε L ◦ s R ) = S ◦ s R ❡ t L ◦ (ε L ◦ s R ) = s R = S ◦ t R .

  ❚❛♠❜é♠✱ ♣❛r❛ t♦❞♦ h ∈ H✱ t❡♠♦s ε L ◦ S(h) = ε L (S(t L (ε L (h ))h ))

  (2) (1)

  = ε L (S(h )s L ◦ ε L (h ))

  (1) (2)

  = ε L (S(h )h )

  (1) (2)

  = ε L (s R ◦ ε R (h)) = (ε L ◦ s R ) ◦ ε R (h). L ◦ S = (S ⊗ L S) ◦ ∆ . cop ❋❛❧t❛ ♠♦str❛r♠♦s ❛♣❡♥❛s q✉❡ ∆ ❉❡ ❢❛t♦✱ ♣❛r❛ R t♦❞♦ h ∈ H✱ t❡♠♦s ∆ L (S(h))

  = ∆ L (S(t L (ε L (h ))h ))

  (2) (1)

  = ∆ L (S(h )s L (ε L (h )))

  (1) (2)

  = S(h ) s L (ε L (h )) ⊗ L S(h )

  (1) (1) (2) (1) (2) (1) (2)

  = S(h ) h S(h ) ⊗ L S(h )

  (1) (1) (2) (2) (1) (2) (1) (1) (2) (1)

  = S(h ) h S(h ) ⊗ L S(h )

  (1) (1) (2) (1) (2) (1) (1) (1) (2) (1)

  = S(h ) t L (ε L (h ))h S(h ) ⊗ L S(h )

  

(1) (1) (2)(2) (2)(1) (1) (2)

(1) (1) (2) (1) (1)

  = S(h ) h S(h ) ⊗ L S(h ) s L (ε L (h ))

  (1) (1) (2)(1) (1) (2) (2)(2)

(1) (1) (2) (1) (1) (1) (1) (2)

  = S(h ) h S(h ) ⊗ L S(h ) h S(h )

  

(1) (1) (2)(1) (1) (2) (2)(2) (2)(2)

(1) (1) (1) (2) (1) (1) (1) (1) (2)

  = S(h ) h S(h ) ⊗ L S(h ) h S(h )

  

(1) (1) (2) (1) (1) (2) (2) (2) (2)

(1)(1) (1)(1) (2) (1)(1) (1)(1) (1)(2)

  = S(h ) h S(h )⊗ L S(h ) h S(h )

  

(1) (1) (2)(1) (1) (2) (2)(2)

(1)(1) (1)(1) (2) (1)(1) (1)(1) (1)(2)

  = (S(h )h ) S(h ) ⊗ L (S(h )h ) S(h )

  

(1) (2) (1) (1) (2) (2)

(1)(1) (2) (1)(1) (1)(2)

  = (s R ◦ ε R (h )) S(h ) ⊗ L (s R ◦ ε R (h )) S(h )

  (1) (2) (2) (1)(1) (1)(2)

  = S(h ) ⊗ L s R ◦ ε R (h )S(h )

  (2) (1)(2) (1)(1)

  = S(h ) ⊗ L S(h t R (ε R (h )))

  (2) (1)

  = S(h ) ⊗ L S(h )

  (2) (1)

  = (S ⊗ L S)(h ⊗ L h ) cop = (S ⊗ L S)∆ (h). R

  ✸✳✸ ◆♦çõ❡s ❆❧t❡r♥❛t✐✈❛s

  ❊①✐st❡ ❝♦♥s❡♥s♦ ♥❛ ❧✐t❡r❛t✉r❛ ❞❡ q✉❡ ❛ ❡str✉t✉r❛ q✉❡ ♠❡❧❤♦r s✉❜s✲ t✐t✉✐ ✉♠❛ ❜✐á❧❣❡❜r❛✱ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❜❛s❡ ♥ã♦✲❝♦♠✉t❛t✐✈❛✱ é ✉♠ ❜✐❛❧❣❡❜ró✐❞❡✳ ❏á ♣❛r❛ s✉❜st✐t✉✐r ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢✱ ❡①✐st❡♠ ❛❧❣✉♠❛s ❞✐s❝✉ssõ❡s ❛❝❡r❝❛ ❞❡ q✉❛❧ ❡str✉t✉r❛ ♠❡❧❤♦r ❣❡♥❡r❛❧✐③❛✳ ◆❡st❛ s❡çã♦ ❞❡✜♥✐♠♦s ❞✉❛s ♥♦çõ❡s✱ ❞✐❢❡r❡♥t❡s ❞❛ ♥♦çã♦ ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✱ q✉❡ t❛♠❜é♠ ❣❡♥❡r❛❧✐③❛♠ ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢✳

  ✸✳✸✳✶ ❍♦♣❢ ❆❧❣❡❜ró✐❞❡s ❞❡ ▲✉

  ❆ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✱ ❝✐t❛❞❛ ❡♠ ✭❬✶✽❪✱ ❉❡✜♥✐çã♦ ✹✳✶✮✱ ✉s❛ ❛♣❡♥❛s ✉♠❛ ❡str✉t✉r❛ ❞❡ ❜✐❛❧❣❡❜ró✐❞❡✱ ❡♥q✉❛♥t♦ q✉❡ ♥❛ ❞❡✜♥✐çã♦ ❞❡ ❍♦♣❢ ❛❧❣❡✲ ❜ró✐❞❡✱ ✉s❛✲s❡ ❞✉❛s ❡str✉t✉r❛s✳ ❊♠❜♦r❛ ♦ ♣r✐♠❡✐r♦ ❛①✐♦♠❛ ❞❛ ❛♥tí♣♦❞❛ ♥❛ ❞❡✜♥✐çã♦ ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ ♣♦❞❡ s❡r ❢♦r♠✉❧❛❞♦ t❛♠❜é♠ ♥❡st❡ ❝❛s♦✱ ❛❧❣✉♠❛s ❝♦♥❞✐çõ❡s ❛❞✐❝✐♦♥❛✐s sã♦ ♥❡❝❡ssár✐❛s ♣❛r❛ ❢♦r♠✉❧❛r ♦ s❡❣✉♥❞♦ ❛①✐♦♠❛✳ ❉❡✜♥✐çã♦ ✸✳✶✶ ❙❡❥❛ B ✉♠ ❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛ s♦❜r❡ L✱ ✉♠❛ k✲ á❧❣❡❜r❛✳ ❉✐③❡♠♦s q✉❡ B é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ ❞❡ ▲✉ s❡ ❡①✐st❡♠ S : B −→ B

  ❡ ✉♠❛ s❡çã♦ ❞❡ k✲♠ó❞✉❧♦s ξ✱ ♣❛r❛ ♦ ❡♣✐♠♦r✜s♠♦ ❝❛♥ô♥✐❝♦ π : B ⊗ k B −→ B ⊗ L B B ⊗ B

  ✱ ♦✉ s❡❥❛✱ π ◦ ξ = Id L ✱ t❛✐s q✉❡✱ ♦s s❡❣✉✐♥t❡s ❛①✐♦♠❛s sã♦ s❛t✐s❢❡✐t♦s✿

  ✭✐✮ S ◦ t = s❀ B ◦ (S ⊗ L

  B) ◦ ∆ = t ◦ ε ◦ S ✭✐✐✮ µ ❀ B ◦ (B ⊗ L S) ◦ ξ ◦ ∆ = s ◦ ε

  ✭✐✐✐✮ µ ✳ ◆❡♥❤✉♠❛ ❞❛s ♥♦çõ❡s✱ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ ❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ ❞❡ ▲✉✱ é

  ♠❛✐s ❣❡r❛❧ ❞♦ q✉❡ ❛ ♦✉tr❛✳ ❉❡ ❢❛t♦✱ ✉♠ ❡①❡♠♣❧♦ ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ q✉❡ ♥ã♦ é ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ ❞❡ ▲✉ é ❝♦♥str✉í❞♦ ❛ s❡❣✉✐r✳

  ❊①❡♠♣❧♦ ✸✳✶✷ ❙❡❥❛ k ✉♠ ❝♦r♣♦ ❝♦♠ ❝❛r❛❝t❡ríst✐❝❛ ❞✐❢❡r❡♥t❡ ❞❡ ❞♦✐s✳ ❈♦♥s✐❞❡r❡ ❛ ❜✐á❧❣❡❜r❛ ❞❡ ❣r✉♣♦ kZ

  2 ✱ ❝♦♠ ❡str✉t✉r❛ ❞❡ k✲❝♦❛♥❡❧ ❞❛❞❛ L (t) = t ⊗ t L (t) = 1 = 1

  2

  ♣♦r ∆ ❡ ε ✱ ❡♠ q✉❡ t ❡ ❝♦♠♦ ✉♠ k✲❜✐❛❧❣❡❜ró✐❞❡ e , η, η)

  à ❡sq✉❡r❞❛✱ ❝♦♠ ❡str✉t✉r❛ ❞❡ k ✲❛♥❡❧ (kZ

  2 ✱ ❡♠ q✉❡ η : k −→

  kZ λ 7−→ λ1

  2 ✳ P♦❞❡♠♦s t❛♠❜é♠✱ ♠✉♥✐r kZ 2 ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ e

  k ✲❜✐❛❧❣❡❜ró✐❞❡ à ❞✐r❡✐t❛✱ ❝♦♠ ❛ ♠❡s♠❛ ❡str✉t✉r❛ ❞❡ k ✲❛♥❡❧ ❡ ❝♦♠ ❡s✲

  2 ✱ ❡♠ q✉❡ ∆ ✱ ∆

  , ∆ R , ε R ) R (t) = −t ⊗ t R (1) = tr✉t✉r❛ ❞❡ k✲❝♦á❧❣❡❜r❛ (kZ

  1 ⊗ 1 R (t) = −1 R (1) = 1 ❡ ε ❡ ε ✳ ❊st❛s ❡str✉t✉r❛s ❥✉♥t♦ ❝♦♠ ❛ ❛♥tí♣♦❞❛

  S(t) := −t ❡ S(1) := 1✱ ❝♦♥st✐t✉❡♠ ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✱ ♣♦ré♠ ♥ã♦ s❛t✐s❢❛③ ♦s ❛①✐♦♠❛s ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ ❞❡ ▲✉✳

  ❉❡ ❢❛t♦✱ ♠♦str❡♠♦s q✉❡ ♦s ❛①✐♦♠❛s ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ sã♦ s❛t✐s❢❡✐t♦s✳ ❈❧❛r♦ q✉❡ ❛ ❝♦♥❞✐çã♦ (i) é tr✐✈✐❛❧♠❡♥t❡ s❛t✐s❢❡✐t❛✳ ❆❣♦r❛ ♥♦t❡ q✉❡

  (∆ L ⊗ k kZ )∆ R (t) = (∆ L ⊗ k kZ )(−t ⊗ t)

  2

  2

  = −t ⊗ t ⊗ t, ♣♦r ♦✉tr♦ ❧❛❞♦✱

  (kZ ⊗ k ∆ R )∆ L (t) = (kZ ⊗ k ∆ R )(t ⊗ t)

  2

  2

  = t ⊗ −t ⊗ t R ⊗ k kZ = −t ⊗ t ⊗ t.

  2 )∆ L = (kZ 2 ⊗ k ∆ L )∆ R .

  ❆♥❛❧♦❣❛♠❡♥t❡✱ ♠♦str❛✲s❡ (∆ ❆❣♦r❛ t❡♠♦s µ(kZ ⊗ k S)∆ R (t) = −tS(t) = t t

  2

  2

  = t = 1 = ε L (t)1 = η ◦ ε L (t), t❛♠❜é♠ t❡♠♦s

  µ(S ⊗ k kZ

  2 )∆ L (t) = S(t)t = −t t

  2

  = −t = ε R (t)1 = η ◦ ε R (t). P❡❧❛ k✲❧✐♥❡❛r✐❞❛❞❡ ❞❡ S✱ ♦ ❛①✐♦♠❛ (iii) é s❛t✐s❢❡✐t♦✳ ❆❣♦r❛ ♣❡r❝❡❜❛ q✉❡✱ ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛

  π : kZ ⊗ k kZ −→ kZ ⊗ L kZ

  2

  2

  2

  2

  é ❛ ✐❞❡♥t✐❞❛❞❡✱ ❞❡st❛ ❢♦r♠❛✱ ❛ ú♥✐❝❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ♣❛r❛ ❛ s❡çã♦ ξ : kZ ⊗ L kZ −→ kZ ⊗ k kZ ,

  2

  2

  2

  2

  é ❛ ✐❞❡♥t✐❞❛❞❡✳ P♦rt❛♥t♦✱ µ(kZ

  2 ⊗ k S)∆ L (t) = −1 6= 1 = η ◦ ε L (t), L , S)

  ✐st♦ ❝♦♥tr❛❞✐③ ♦ ❢❛t♦ q✉❡ (H é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ ❞❡ ▲✉✳ R

  ✸✳✸✳✷ × ✲❍♦♣❢ ➪❧❣❡❜r❛

  ❙❛❜❡♠♦s ❞❛ t❡♦r✐❛ ❞❡ á❧❣❡❜r❛s ❞❡ ❍♦♣❢ q✉❡ ♦s ❝♦✐♥✈❛r✐❛♥t❡s ❞♦ ❝♦✲ ♠ó❞✉❧♦ r❡❣✉❧❛r ❞❡ ✉♠❛ ❜✐á❧❣❡❜r❛ H✱ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k✱ sã♦ ♣r❡❝✐s❛♠❡♥t❡ ♦s ♠ú❧t✐♣❧♦s ❡s❝❛❧❛r❡s ❞❛ ✉♥✐❞❛❞❡✳ H é ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ H é ✉♠❛ H✲❡①t❡♥sã♦ ●❛❧♦✐s ❞❡ k✱ ♦✉ s❡❥❛✱ ❛ K H −→ H ⊗ k

  H, h ⊗ k 7−→ h ⊗ h k ❛♣❧✐❝❛çã♦ Can : H ⊗ (1) (2) ✱ é ❜✐❥❡t✐✈❛✳ ❉❡ ❢❛t♦✱ ❡①✐st❡ ✉♠ ✐s♦♠♦r✜s♠♦ ❡♠ q✉❡ ❛ ♣r✐♠❡✐r❛ á❧❣❡❜r❛ é ♣♦r ❝♦♠♣♦s✐çã♦ ❡ ❛ s❡❣✉♥❞❛ ♣♦r ❝♦♥✈♦❧✉çã♦ H

  Hom H (H ⊗ k

  H, H ⊗ k H) ∼ k (H, H).

  = Hom ❚❛❧ ✐s♦♠♦r✜s♠♦ é ❞❛❞♦ ♣♦r H c

  ( ) : Hom H (H ⊗ k

  H, H ⊗ k

  H) −→ Hom k (H, H) b F 7−→ F ,

  F (h) = (ε ⊗ H)F (h ⊗ 1 H ) t❛❧ q✉❡ b ✳ ❈♦♠ ✐♥✈❡rs❛ ❞❡✜♥✐❞❛ ♣♦r H f ( ) : Hom k (H, H) −→ Hom H (H ⊗ k

  H, H ⊗ k

  H) e f 7−→ f , f (h⊗k) = h ⊗f (h )k t❛❧ q✉❡ e (1) (2) ✱ ♣❛r❛ q✉❛✐sq✉❡r h, k ∈ H✳ ❊st❡ ✐s♦♠♦r✲ ✜s♠♦ r❡❧❛❝✐♦♥❛ ❛ ✐♥✈❡rs❛ ❞❛ ❝❛♥ô♥✐❝❛ ❝♦♠ ❛ ❛♥tí♣♦❞❛✳ ▼♦t✐✈❛❞♦ ♣♦r ❡st❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ á❧❣❡❜r❛s ❞❡ ❍♦♣❢✱ ❙❝❤❛✉❡♥❜✉r❣ ❡♠ ❬✷✺❪ ♣r♦♣ôs ❛ R ❞❡✜♥✐çã♦ ❞❡ × ✲❍♦♣❢ á❧❣❡❜r❛✳ P❛r❛ ❡♥t❡♥❞❡r♠♦s ❡ss❛ ❞❡✜♥✐çã♦✱ ♣r❡❝✐✲ s❛♠♦s ❞❡ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❞❛ t❡♦r✐❛ ❞❡ ❝❛t❡❣♦r✐❛s✳ ❉❡✜♥✐çã♦ ✸✳✶✸ ❙❡❥❛♠ C ❡ D ❝❛t❡❣♦r✐❛s✳ ❯♠❛ ❛❞❥✉♥çã♦ ❞❡ C ❛ D é ✉♠❛ tr✐♣❧❛ (F, G, φ)✱ ❡♠ q✉❡ F : C −→ D ❡ G : D −→ C sã♦ ❢✉♥t♦✲ XY : Hom (F (X), Y ) −→ Hom (X, G(Y )) D C r❡s ❡ φ

  é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s✱ ♣❛r❛ q✉❛✐sq✉❡r X ∈ C ❡ Y ∈ D✳ ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ F é ❛❞❥✉♥t♦ à ❡sq✉❡r❞❛ ❞❡ G ❡ q✉❡ G é ❛❞❥✉♥t♦ à ❞✐r❡✐t❛ ❞❡ F ✳ ❉❡✜♥✐çã♦ ✸✳✶✹ ❙❡❥❛ C ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ P❛r❛ Y ∈ C✱ s❡ ♦ ❢✉♥t♦r − ⊗ Y : C −→ C é ❛❞❥✉♥t♦ à ❡sq✉❡r❞❛✱ ❞❡♥♦t❛♠♦s s❡✉ ❛❞❥✉♥t♦ à C

  (Y, −) : C −→ C ❞✐r❡✐t❛ ♣♦r hom ❡ ❝❤❛♠❛♠♦s ❞❡ ❤♦♠✲❢✉♥t♦r ✐♥t❡r♥♦ à ❞✐r❡✐t❛✳ ❙❡ ♦ ❢✉♥t♦r − ⊗ Y é ❛❞❥✉♥çã♦ à ❡sq✉❡r❞❛ ♣❛r❛ t♦❞♦ Y ∈ C✱ ❡♥tã♦ ❞✐③❡♠♦s q✉❡ C é ❢❡❝❤❛❞❛ à ❞✐r❡✐t❛✳

  ❙❡❥❛♠ X, Y ♦❜❥❡t♦s ❡♠ C✳ P♦r r❡s✉❧t❛❞♦s ❞❡ ❢✉♥t♦r❡s ❛❞❥✉♥t♦s✱ ✉♠ C (Y, −)

  ❤♦♠✲❢✉♥t♦r ✐♥t❡r♥♦ à ❞✐r❡✐t❛ hom ✈❡♠ ❝♦♠ ✉♠ ♠♦r✜s♠♦ ❛❞❥✉♥✲ C (Y, X) ⊗ Y −→ X

  çã♦ ev : hom ✱ ❝♦♠ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧✿ C (Y, X)

  P❛r❛ ❝❛❞❛ Z ∈ C ❡ e : Z ⊗Y −→ X✱ ❡①✐st❡ ú♥✐❝♦ f : Z −→ hom ✱ t❛❧ q✉❡ e = ev ◦ (f ⊗ Y )✳ ❚❛❧ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧✱ é ✈á❧✐❞❛ ♣❛r❛ q✉❛❧✲ q✉❡r ❛❞❥✉♥çã♦ (F, G, φ)✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ❢❛t♦✱ ♣♦❞❡ s❡r ❡♥❝♦♥✲ tr❛❞❛ ❡♠ ✭❬✷✾❪✱ ❚❡♦r❡♠❛ ✷✳✷✽✮✳

  ❙❡❥❛ R ✉♠❛ á❧❣❡❜r❛ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ k✳ ❈♦♥s✐❞❡r❡ ❛ ❝❛✲ e R e M t❡❣♦r✐❛ ❞♦s R ✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ ✱ ❡st❛ ❝❛t❡❣♦r✐❛ ♣♦❞❡ s❡r ✈✐st❛ R e M ❝♦♠♦ ❛ ❝❛t❡❣♦r✐❛ ❞♦s R✲❜✐♠ó❞✉❧♦s✱ ♣♦✐s s❡ M ∈ ✱ ♣❛r❛ q✉❛✐sq✉❡r

  ′ ′ ′ e M r, r ∈ R = (r⊗r )m R

  ❡ m ∈ M✱ ❞❡✜♥✐♠♦s r·m·r ✳ ❊♥tã♦ é ♠♦♥♦✐❞❛❧ R ❝♦♠ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ⊗ ❡ ✉♥✐❞❛❞❡ ♠♦♥♦✐❞❛❧ R✳ ➱ ♣♦ssí✈❡❧ ♠♦str❛r R , ⊗ R ) e M q✉❡ ❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ( é ❢❡❝❤❛❞❛✱ ❝♦♠ ❤♦♠✲❢✉♥t♦r ✐♥t❡r♥♦ M op e M hom (N, P ) = R Hom(N, P ) R Re ✱ ♣❛r❛ q✉❛✐sq✉❡r N, P ∈ ✱ ❡♠ q✉❡ e

  ❛ ❡str✉t✉r❛ ❞❡ R ✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ é ❞❛❞❛ ♣♦r

  ′ ′

  ((r ⊗ r)f )(n) = r f (rn),

  ′ op

  ∈ R R Hom(N, P ) ♣❛r❛ q✉❛✐sq✉❡r r, r ✱ f ∈ ❡ n ∈ N. k (V, W ) ❙❡ H é ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢ s♦❜r❡ k✱ ❡♥tã♦ ♦ k✲♠ó❞✉❧♦ Hom

  ❞❡ ♠♦r✜s♠♦s k✲❧✐♥❡❛r❡s✱ ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ❝❛♥ô♥✐❝❛ ❞❡ H✲♠ó❞✉❧♦ f (S(h )v) à ❡sq✉❡r❞❛✱ ❞❡✜♥✐❞❛ ♣♦r (hf)(v) = h (1) (2) ✱ ♣❛r❛ q✉❛✐sq✉❡r h ∈ H k (V, W ) k (V, W )

  ✱ f ∈ Hom ❡ v ∈ V ✳ ❈♦♠ ❡st❛ ❡str✉t✉r❛✱ Hom ❞❡✜♥❡ ✉♠ ❤♦♠✲❢✉♥t♦r ✐♥t❡r♥♦ ♥❛ ❝❛t❡❣♦r✐❛ ❞♦s H✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ❆❣♦r❛✱ ❡♠ ❝♦♥tr❛♣❛rt✐❞❛✱ ✉♠❛ ❜✐á❧❣❡❜r❛ ♥ã♦ ♣r❡❝✐s❛ s❡r ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢ ♣❛r❛ q✉❡ s✉❛ ❝❛t❡❣♦r✐❛ ❞❡ ♠ó❞✉❧♦s s❡❥❛ ❢❡❝❤❛❞❛✳ P♦ré♠✱ ❛ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ♠♦str❛rá q✉❡ ❛ ❝❛t❡❣♦r✐❛ ❞❡ ♠ó❞✉❧♦s s♦❜r❡ ✉♠❛ ❜✐á❧❣❡❜r❛ B

  ♦✉ ❛té ♠❡s♠♦ s♦❜r❡ ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛ é ❢❡❝❤❛❞❛✳ Pr♦♣♦s✐çã♦ ✸✳✶✺ ❙❡❥❛ B ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛✳ ❊♥tã♦ ❛ ❝❛✲ B M t❡❣♦r✐❛ ✱ ❞♦s B✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✱ é ❢❡❝❤❛❞❛ à ❞✐r❡✐t❛ ❝♦♠ ❤♦♠✲ B M

  ❢✉♥t♦r ✐♥t❡r♥♦ à ❞✐r❡✐t❛ ❞❛❞♦✱ ♣❛r❛ q✉❛✐sq✉❡r N, P ∈ ✱ ♣♦r M R N hom (N, P ) = B Hom(B ⊗ R N, P ), B ❡♠ q✉❡ B⊗ é ✉♠ B✲❜✐♠ó❞✉❧♦ ❝♦♠ ❡str✉t✉r❛ ❞❡ B✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛

  ′

  ∈ B ❞❛❞❛✱ ♣❛r❛ q✉❛✐sq✉❡r b, b ❡ n ∈ N✱ ♣♦r

  ′ ′

  b ⊲ (b ⊗ R n) = b b ⊗ R b ⊲ n

  (1) (2)

  ❡ ❡str✉t✉r❛ ❞❡ B✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❞❛❞❛ ♣♦r

  ′ ′ (b ⊗ R n) ⊳ b = b b ⊗ R n. B M

  ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r♦✱ ♥♦t❡ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r M, N✱ P ∈ ✱ m ∈ M ❡ n ∈ N t❡♠♦s q✉❡

  ϕ : M ⊗ R N −→ (B ⊗ R N ) ⊗ B M m ⊗ R n 7−→ (1 B ⊗ R n) ⊗ B m, é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ B✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M

  ✱ n ∈ N ❡ b ∈ B✱ ❞❡✜♥❛

  −1

  ϕ : (B ⊗ R N ) ⊗ B M −→ M ⊗ R N (b ⊗ R n) ⊗ B m 7−→ b ⊲ m ⊗ R n. ❆❣♦r❛✱ ❧❡♠❜r❡ q✉❡ ❛ ❡str✉t✉r❛ ❞❡ R✲❜✐♠ó❞✉❧♦ ❡♠ ✉♠ B✲♠ó❞✉❧♦ à ❡s✲

  ′

  ∈ R q✉❡r❞❛ q✉❛❧q✉❡r M✱ é ❞❛❞❛✱ ♣❛r❛ q✉❛✐sq✉❡r r, r ❡ m ∈ M✱ ♣♦r

  

′ ′

r · m · r = s(r)t(r ) ⊲ m.

  ▼♦str❡♠♦s q✉❡ ϕ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ ❉❡ ❢❛t♦✱ ϕ é R✲❜❛❧❛♥❝❡❛❞❛✱ ϕ(m · r, n) = (1 B ⊗ R n) ⊗ m · r

  = (1 B ⊗ R n) ⊗ t(r) ⊲ m = (1 B ⊗ R n) ⊳ t(r) ⊗ m = (t(r)1 B ⊗ R n) ⊗ m = (1 B · r ⊗ R n) ⊗ m = (1 B ⊗ R r · n) ⊗ m = ϕ(m, r · n). R N −→ (B ⊗ R N ) ⊗ B M

  P♦rt❛♥t♦✱ ❡①✐st❡ ú♥✐❝❛ ϕ : M ⊗ ✱ q✉❡ s❛t✐s❢❛③ ϕ(m ⊗ R n) = (1 B ⊗ R n) ⊗ B m

  ✱ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M ❡ n ∈ N✳ ▼♦str❡♠♦s q✉❡ ϕ é ♠♦r✜s♠♦ ❞❡ B✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M✱ n ∈ N ❡ b ∈ B✱ t❡♠♦s R R

  ϕ(b ⊲ (m ⊗ n)) = ϕ(b ⊲ m ⊗ b ⊲ n)

  (1) (2)

  = (1 B ⊗ R b ⊲ n) ⊗ B b ⊲ m

  (2) (1)

  = (1 B ⊗ R b ⊲ n) ⊳ b ⊗ B m

  (2) (1)

  = (b ⊗ R b ⊲ n) ⊗ B m

  (1) (2)

  = b ⊲ (1 B ⊗ R n) ⊗ B m

  = b ⊲ ϕ(m ⊗ R n).

  −1

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ϕ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r

  ′ −1

  r ∈ R ∈ B ✱ m ∈ M✱ n ∈ N ❡ b, b ✱ t❡♠♦s q✉❡ ϕ é R✲❜❛❧❛♥❝❡❛❞❛ ❡

  B ✲❜❛❧❛♥❝❡❛❞❛

  −1

  ϕ ((b · r, n), m) = (b · r) ⊲ m ⊗ R n = t(r)b ⊲ m ⊗ R n = t(r) ⊲ (b ⊲ m) ⊗ R n = (b ⊲ m) · r ⊗ R n = (b ⊲ m) ⊗ R r · n

  −1

  = ϕ ((b, r · n), m),

  −1 ′ ′

  ϕ ((b, n), b ⊲ m) = bb ⊲ m ⊗ R n

  −1 ′

  = ϕ ((bb ⊗ R n), m)

  −1 ′ = ϕ ((b ⊗ R n) ⊳ b , m).

  P♦rt❛♥t♦✱ t❡♠♦s

  −1 −1 R B R B

  (ϕ ◦ ϕ )((b ⊗ n) ⊗ m) = ϕ(ϕ ((b ⊗ n) ⊗ m)) = ϕ(b ⊲ m ⊗ R n) = (1 B ⊗ R n) ⊗ B b ⊲ m = (1 B ⊗ R n) ⊳ b ⊗ B m = (b ⊗ R n) ⊗ B m,

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s

  −1 −1

  (ϕ ◦ ϕ)(m ⊗ R n) = ϕ ((1 B ⊗ R n) ⊗ B m) = 1 B ⊲ m ⊗ R n = m ⊗ R n. B M

  ❉❡✜♥❛ ❛❣♦r❛✱ ♣❛r❛ q✉❛✐sq✉❡r M, N✱ P ∈ ✱ ❛ ❛♣❧✐❝❛çã♦ θ : B Hom(M ⊗ R N, P ) −→ B Hom((B ⊗ R N ) ⊗ B M, P )

  −1 f 7−→ θ(f ) = f ◦ ϕ .

  ❈❧❛r♦ q✉❡ θ(f) é ♠♦r✜s♠♦ ❞❡ B✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✱ ♣♦✐s é ❛ ❝♦♠✲

  −1

  (g) = g ◦ ϕ B Hom((B ⊗ R N ) ⊗ B M ) ♣♦s✐çã♦ ❞❡ ♦✉tr♦s ❞♦✐s✳ ❆❣♦r❛ ❞❡✜♥❛ θ ✱ ♣❛r❛ t♦❞♦ g ∈ ✳ ❉❡st❛ ❢♦r♠❛✱ t❡♠♦s

  

−1 −1 −1 −1

θ(θ (g)) = θ (g) ◦ ϕ = g ◦ ϕ ◦ ϕ = g.

  −1

  (θ(f )) = f ❆♥❛❧♦❣❛♠❡♥t❡✱ θ ✳ ❙❡❣✉❡ ♣♦rt❛♥t♦✱ ♦ ✐s♦♠♦r✜s♠♦ ❞❡ B✲ ♠ó❞✉❧♦s à ❡sq✉❡r❞❛ B Hom(M ⊗ R N, P ) ∼ B Hom((B ⊗ R N ) ⊗ B M, P ).

  = B M ▼♦str❡♠♦s ❛❣♦r❛ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r M, N✱ P ∈ ❡ m ∈ M✱ n ∈ N✱ b ∈ B

  ✱ ❛ ❛♣❧✐❝❛çã♦ c ( ) : B Hom((B ⊗ R N ) ⊗ B M, P )−→ B Hom(M, B Hom(B ⊗ R N, P )) b

  F 7−→ F , ❡♠ q✉❡

  F : b M −→ B Hom(B ⊗ R N, P ) b m 7−→ F (m),

  F (m)(b ⊗ R n) = F ((b ⊗ R n) ⊗ B m) t❛❧ q✉❡ b ✱ é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ B

  ✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✱ ❝♦♠ ❛ ❡str✉t✉r❛ ❞❡ B✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ ❡♠ B Hom(B ⊗ R N, P ) ⊗ R n) = f (b b ⊗ R n) ′ ′ ❞❛❞❛ ♣♦r (b ⊲ f)(b ✱ ♣❛r❛

  ′

  ∈ B q✉❛✐sq✉❡r b, b ❡ n ∈ N✳ ❙❡❣✉❡ ❞❛ ♠❛♥❡✐r❛ ❝♦♠♦ ❢♦✐ ❞❡✜♥✐❞♦✱ F (m) q✉❡ ♣❛r❛ t♦❞♦ m ∈ M✱ b é R✲❜❛❧❛♥❝❡❛❞♦✳ ▼♦str❡♠♦s ❡♥tã♦ q✉❡ é

  ♠♦r✜s♠♦ ❞❡ B✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M✱

  ′

  n ∈ N ∈ B ❡ b, b ✱ t❡♠♦s

  ′ ′

  b F (m)(b ⊲ (b ⊗ R n)) = F (b ⊲ (b ⊗ R n) ⊗ B m)

  ′

  = b ⊲ F ((b ⊗ R n) ⊗ B m)

  ′

  = b ⊲ b F (m)(b ⊗ R n) F

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ b é ♠♦r✜s♠♦ ❞❡ B✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱ t❡♠♦s

  ′ ′

  b F (b ⊲ m)(b ⊗ R n) = F ((b ⊗ R n) ⊗ B b ⊲ m)

  ′ R B

  = F ((b ⊗ n) ⊳ b ⊗ m)

  ′

  = F ((b b ⊗ R n) ⊗ B m)

  ′

  = b F (m)(b b ⊗ R n)

  ′

  = b ⊲ b F (m)(b ⊗ R n)

  ′ = (b ⊲ b F )(m)(b ⊗ R n).

  ❆❣♦r❛ ❞❡✜♥❛ f ( ) : B Hom(M, B Hom(B ⊗ R N, P ))−→ B Hom((B ⊗ R N ) ⊗ B M, P ) e

  G 7−→

  G, G((b ⊗ R n) ⊗ B m) = G(m)(b ⊗ R n) t❛❧ q✉❡ e ✱ ♣❛r❛ q✉❛✐sq✉❡r b ∈ B✱ m ∈ M ( ) ( )

  ❡ n ∈ N✳ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❛ c ♠♦str❛✲s❡ q✉❡ f ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ ❉❡st❛ ❢♦r♠❛✱ ♣❛r❛ q✉❛✐sq✉❡r m ∈ M✱ n ∈ N✱ b ∈ B✱ B Hom(M, B Hom(B ⊗ R N, P )) B Hom((B ⊗ R N ) ⊗ B M, P ) ❡ t❛♠❜é♠ ♣❛r❛ q✉❛✐sq✉❡r G ∈ ❡ F ∈

  ✱ t❡♠♦s G(m)(b ⊗ be R n) = e G((b ⊗ R n) ⊗ B m)

  = G(m)(b ⊗ R n), G = G s❡❣✉❡ q✉❡ be ✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s

  F ((b ⊗ eb R n) ⊗ B m) = b F (m)(b ⊗ R n) = F ((b ⊗ R n) ⊗ B m),

  F = F s❡❣✉❡ ♣♦rt❛♥t♦✱ eb ✳ ❈♦♥❝❧✉í♠♦s ❛ss✐♠✱ q✉❡ B Hom((B ⊗ R N ) ⊗ B M, P ) ∼ = B Hom(M, B Hom(B ⊗ R N, P )).

  ❖✉ s❡❥❛✱ B Hom(M ⊗ R N, P ) ∼ = B Hom(M, B Hom(B ⊗ R N, P )).

  P❛r❛ s❛❧✈❛r ❛ ✐❞❡✐❛ ❞❡ q✉❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ♦ ❢❛t♦ ❞❡ s✉❛ ❝❛t❡❣♦r✐❛ ❞❡ ♠ó❞✉❧♦s s❡r ❢❡❝❤❛❞❛✱ ♣r❡❝✐s❛♠♦s ♦❜✲ s❡r✈❛r q✉❡ ♦ ❤♦♠✲❢✉♥t♦r ✐♥t❡r♥♦ ♥❛ ❝❛t❡❣♦r✐❛ ❞❡ ♠ó❞✉❧♦s s♦❜r❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢✱ ❡stá r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ♦ ❤♦♠✲❢✉♥t♦r ✐♥t❡r♥♦ ♥❛ ❝❛t❡❣♦✲ r✐❛ s✉❜❥❛❝❡♥t❡ ❞❡ k✲♠ó❞✉❧♦s✳ ❊st❛ r❡❧❛çã♦ s❡ ❞á ✈✐❛ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ H −→ k M M q✉❡✱ ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ ♣r❡s❡r✈❛ ❤♦♠✲❢✉♥t♦r❡s ✐♥t❡r✲

  ♥♦s✱ ♥♦ s❡♥t✐❞♦ ❞❛ ♣ró①✐♠❛ ❞❡✜♥✐çã♦✳ ❉❡✜♥✐çã♦ ✸✳✶✻ ❙❡❥❛ F : C −→ D ✉♠ ❢✉♥t♦r ♠♦♥♦✐❞❛❧ ❡♥tr❡ ❝❛t❡❣♦r✐❛s ♠♦♥♦✐❞❛✐s ❢❡❝❤❛❞❛s à ❞✐r❡✐t❛✳ P❛r❛ X, Y ∈ C✱ ❝♦♥s✐❞❡r❡ C D

  ξ : F(hom (Y, X)) −→ hom (F(Y ), F(X)), ♦ ú♥✐❝♦ ♠♦r✜s♠♦ ♣❛r❛ ♦ q✉❛❧ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ ❝♦♠✉t❛ ξ

  ⊗F(Y )

  F C // D (hom (Y, X)) ⊗ F(Y ) hom (F(Y ), F(X)) ⊗ F(Y )

  ∼

= ev

  F C (hom (Y, X) ⊗ Y ) // F(X). F

  (ev)

  ❉✐③❡♠♦s q✉❡ F ♣r❡s❡r✈❛ ❤♦♠✲❢✉♥t♦r❡s ✐♥t❡r♥♦s s❡ ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈ C

  ✱ ξ é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❆♣r❡s❡♥t❛♠♦s ❛ s❡❣✉✐r ♦ ▲❡♠❛ ❞❡ ❨♦♥❡❞❛✱ q✉❡ s❡rá ✉s❛❞♦ ♥❛ ♣r♦✈❛

  ❞♦ ♣ró①✐♠♦ t❡♦r❡♠❛✳ P❛r❛ t❛♥t♦✱ ❝♦♥s✐❞❡r❡ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❡ X ✉♠ X : C −→ Set

  ♦❜❥❡t♦ ✜①♦ ❡♠ C✳ ❉❡✜♥✐♠♦s ♦ ❢✉♥t♦r L ✱ ♣❛r❛ t♦❞♦ Y ∈ C✱ X = Hom (X, Y ) C ♣♦r L ✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ ❝❛❞❛ ♠♦r✜s♠♦ α : Y −→ Z✱ ❞❡✜♥✐♠♦s C C

  L X (α) : Hom (X, Y ) −→ Hom (X, Z) f 7−→ α ◦ f. ❉❡st❛ ❢♦r♠❛✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ ❧❡♠❛✳ ▲❡♠❛ ✸✳✶✼ ✭▲❡♠❛ ❞❡ ❨♦♥❡❞❛✮ ❙❡❥❛♠ F : C −→ Set ✉♠ ❢✉♥t♦r ❡

  X ✉♠ ♦❜❥❡t♦ ❡♠ C✳ ❊♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❞❛s tr❛♥s❢♦r♠❛çõ❡s ♥❛t✉r❛✐s

  N at(L X , F ) ❡stá ❡♠ ❜✐❥❡çã♦ ❝♦♠ ♦ ❝♦♥❥✉♥t♦ F (X) ♣❡❧❛ ❢✉♥çã♦

  φ : N at(L X , F ) −→ F (X) X µ 7−→ µ X (I X ), ❡♠ q✉❡ I é ♦ ♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ ❞❡ X✳ Pr♦♣♦s✐çã♦ ✸✳✶✽ ❙❡❥❛♠ C ✉♠❛ ❝❛t❡❣♦r✐❛ ❡ X✱ Y ♦❜❥❡t♦s ❡♠ C✳ ❊♥tã♦ L X Y

  é ✐s♦♠♦r❢♦ ❛ L s❡✱ ❡ s♦♠❡♥t❡ s❡✱ X é ✐s♦♠♦r❢♦ ❛ Y ✳ ❆s ❞❡♠♦♥str❛çõ❡s ❞♦ ▲❡♠❛ ❞❡ ❨♦♥❡❞❛ ❡ ❞❛ ♣r♦♣♦s✐çã♦ ✸✳✶✽✱ ♣♦❞❡♠ s❡r❡♠ ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✷✾❪✱ ▲❡♠❛ 2.23 ❡ Pr♦♣♦s✐çã♦ 2.24✱ r❡s♣❡❝t✐✈❛✲

  ♠❡♥t❡✳ ◆❡st❛s ❝♦♥❞✐çõ❡s✱ ❡st❛♠♦s ♣r♦♥t♦s ♣❛r❛ ✈❡r ❛ ❞❡✜♥✐çã♦ ✭t❡♦r❡♠❛✮ R

  ❞❡ × ✲❍♦♣❢ á❧❣❡❜r❛✳ ❚❡♦r❡♠❛ ✸✳✶✾ ✭❡ ❉❡✜♥✐çã♦✮ ❙❡❥❛ B ✉♠ R✲❜✐❛❧❣❡❜ró✐❞❡ à ❡sq✉❡r❞❛✳ ❊♥tã♦ ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿

  M M e (1) B −→ R

  ❖ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ ♣r❡s❡r✈❛ ❤♦♠✲❢✉♥t♦r❡s ✐♥✲ t❡r♥♦s à ❞✐r❡✐t❛✳ (2)

  ❆ ❛♣❧✐❝❛çã♦ op Can : B ⊗ R B −→ B ⊗ R B op ′ ′ b ⊗ R b 7−→ b ⊗ R b b ,

  (1) (2) R

  é ✉♠❛ ❜✐❥❡çã♦✳ ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ B é ✉♠❛ × ✲❍♦♣❢ á❧✲ ❣❡❜r❛✳

  ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ ✈❛♠♦s ✈❡r q✉❡ Can ❡stá ❜❡♠ ❞❡✜✲ R B op ♥✐❞❛✳ P❛r❛ t❛♥t♦✱ ❛s ❡str✉t✉r❛s ❞❡ B✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ ❡♠ B ⊗ ❡ op op

  ′ ′

  R , c ∈ B ∈ R

  ✲❜✐♠ó❞✉❧♦ ❡♠ B sã♦ ❞❛❞❛s✱ ♣❛r❛ q✉❛✐sq✉❡r b, b ❡ r, r ✱ ♣♦r

  ′ ′ op op

  b ⊲ (b ⊗ R

  c) = bb ⊗ R c ❡

  ′ ′

  r · b · r = t(r)b t(r ), op r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ss✐♠✱ t❡♠♦s q✉❡ Can é R ✲❜❛❧❛♥❝❡❛❞❛✱

  ′ ′

  Can(b · r, b ) = Can(b t(r), b )

  ′

  = b ⊗ R b t(r)b

  (1) (2) ′

  = b ⊗ R b (r · b )

  (1) (2) ′

  = Can(b, r · b ), ✉s❛♠♦s ♥❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♦ s❡❣✉✐♥t❡ ❢❛t♦✱ ∆(bt(r)) = ∆(b)∆(t(r)) = (b ⊗ R b )(1 B ⊗ R t(r)) = b ⊗ R b t(r).

  (1) (2) (1) (2)

  ❚❡♠♦s t❛♠❜é♠ q✉❡ Can é ♠♦r✜s♠♦ ❞❡ B✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱

  

′ ′

op op

  Can(b ⊲ (b ⊗ R

  c)) = Can(bb ⊗ R

  c)

  ′ ′

  = b b ⊗ R b b c

  (1) (2) (1) (2) ′ ′

  = b ⊲ (b ⊗ R b

  c)

  (1) (2) ′ op

  = b ⊲ Can(b ⊗ R c). B M ❆❣♦r❛✱ ♥♦t❡ q✉❡ s❡ Can ❢♦r ✉♠❛ ❜✐❥❡çã♦✱ ❡♥tã♦ ♣❛r❛ N ∈ ❛ ❛♣❧✐✲ ❝❛çã♦ op

  Can N : B ⊗ R N −→ B ⊗ R N op b ⊗ R n 7−→ b ⊗ R b ⊲ n,

  (1) (2)

  t❛♠❜é♠ é ✉♠❛ ❜✐❥❡çã♦✳ ❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❡ ❛s ❜✐❥❡çõ❡s op op φ : B ⊗ R N −→ B ⊗ R B ⊗ B N op op b ⊗ R n 7−→ b ⊗ R

  1 B ⊗ B n, ψ : B ⊗ R B ⊗ B N −→ B ⊗ R N

  ′ ′ B N b ⊗ R b ⊗ B n 7−→ b ⊗ R b ⊲ n.

  ❡ t❛♠❜é♠ Can ⊗ ✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❛r❛ q✉❛✐sq✉❡r b ∈ B ❡ n ∈ N✱ t❡♠♦s op op (ψ ◦ (Can ⊗ B N ) ◦ φ)(b ⊗ R n) = ψ(Can ⊗ B N )(φ(b ⊗ R n))

  = ψ(Can ⊗ B N )(b ⊗ R op

  = (t(r) ⊲ g)(1 B ⊗ R n) = g(t(r) ⊗ R n) = g(1 B · r ⊗ R n) = g(1 B ⊗ R r · n) = e ev(g, r · n).

  (2)

  ⊗ R b

  (1)

  ⊲ n) = g(b

  (2)

  ⊲ g)(1 B ⊗ B b

  

(1)

  ⊲ n) = (b

  (2)

  ⊲ g ⊗ R b

  (1)

  P♦rt❛♥t♦✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ev : B Hom(B ⊗ R N, P ) ⊗ R N −→ P, q✉❡ s❛t✐s❢❛③ ev(g ⊗ R n) = g(1 B ⊗ R n) ✳ ❆❣♦r❛ ♥♦t❡ q✉❡ ev é ♠♦r✜s♠♦ ❞❡ B✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱ t❡♠♦s ev(b ⊲ (g ⊗ R n)) = ev(b

  ❡ ♠♦str❡♠♦s q✉❡ e ev é R✲❜❛❧❛♥❝❡❛❞❛✱ e ev(g · r, n) = e ev(t(r) ⊲ g, n)

  1 B ⊗ B n) = ψ(b

  ✱ n ∈ N ❡ b ∈ B✱ ❞❡✜♥❛ e ev : B Hom(B ⊗ R N, P ) × N −→ P, t❛❧ q✉❡ e ev(g, n) = g(1 B ⊗ R n)

  ✱ ❛ ❛✈❛❧✐❛çã♦ ev : B Hom(B ⊗ R N, P ) ⊗ R N −→ P, é ❞❛❞❛✱ ♣❛r❛ q✉❛✐sq✉❡r g ∈ B Hom(B ⊗ R N, P ) ❡ n ∈ N✱ ♣♦r ev(g ⊗ R n) = g(1 B ⊗ R n). ▼♦str❡♠♦s q✉❡ ev ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r g ∈ B Hom(B ⊗ R N, P )

  ♣❛r❛ ♦ ❤♦♠✲❢✉♥t♦r ✐♥t❡r♥♦ ❡♠ B M

  M ✳ ❱❡r✐✜q✉❡♠♦s q✉❡

  ❙❡❣✉❡ q✉❡ Can N é ❜✐❥❡çã♦✳ ❈♦♥❝❧✉í♠♦s q✉❡ Can é ❜✐❥❡çã♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ Can N é ❜✐❥❡çã♦✳ P♦✐s s❡ Can N é ❜✐❥❡çã♦✱ s❡❣✉❡ q✉❡ Can t❛♠❜é♠ é✱ ❜❛st❛ t♦♠❛r N = B✳ ❆❣♦r❛✱ s❡❥❛♠ N, P ∈ B

  ⊲ n = Can N (b ⊗ R op n).

  (2)

  ⊗ R b

  (1)

  ⊗ B n) = b

  (2)

  ⊗ R b

  (1)

  ⊲ n) = g(b ⊲ (1 B ⊗ R n))

  = b ⊲ g(1 B ⊗ R n) = b ⊲ ev(g ⊗ R n). B R N −→ P M ❆❣♦r❛✱ ♣❛r❛ ❝❛❞❛ Z ∈ ❡ e : Z ⊗ ✱ ❞❡✜♥❛ f : Z −→ B Hom(B ⊗ R N, P ), t❛❧ q✉❡ f (z) : B ⊗ R N −→ P, b ⊗ R n 7−→ e(b ⊲ z ⊗ R n), ♣❛r❛ q✉❛✐sq✉❡r z ∈ Z✱ b ∈ B ❡ n ∈ N✳ ▼♦str❡♠♦s q✉❡ f ❡stá ❜❡♠ B Hom(B ⊗ R N, P ) ∀z ∈ Z ❞❡✜♥✐❞❛✳ P❛r❛ t❛♥t♦✱ ♥♦t❡ q✉❡ f(z) ∈ ✳ ❉❡

  ′

  ∈ B f (z) : B×N −→ P ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r r ∈ R✱ b, b ❡ n ∈ N✱ ❞❡✜♥❛ g f (z)(b, n) = e(b ⊲ z ⊗ R n) f (z) t❛❧ q✉❡ g ✱ ♣❛r❛ z ∈ Z ✜①♦ ❡ ♠♦str❡♠♦s q✉❡ g é R✲❜❛❧❛♥❝❡❛❞❛✱ g f (z)(b · r, n) = g f (z)(t(r)b, n)

  = e(t(r)b ⊲ z ⊗ R n) = e(t(r) ⊲ (b ⊲ z) ⊗ R n) = e((b ⊲ z) · r ⊗ R n) = e((b ⊲ z) ⊗ R r · n) = g f (z)(b, r · n). R N −→ P

  P♦rt❛♥t♦✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ f(z) : B ⊗ ✱ q✉❡ s❛t✐s❢❛③ f(z) = e(b ⊲ z ⊗ R n) ✱ ♣❛r❛ q✉❛✐sq✉❡r b ∈ B ❡ n ∈ N✳ ❆❣♦r❛ ♥♦t❡ q✉❡ f(z) é

  ♠♦r✜s♠♦ ❞❡ B✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱

  ′ ′

  f (z)(b ⊲ (b ⊗ R n)) = f (z)(b b ⊗ R b ⊲ n)

  (1) (2) ′

  = e(b b ⊲ z ⊗ R b ⊲ n)

  (1) (2) ′

  = e(b ⊲ (b ⊲ z) ⊗ R b ⊲ n)

  (1) (2) ′

  = e(b ⊲ ((b ⊲ z) ⊗ R n))

  ′

  = b ⊲ e((b ⊲ z) ⊗ R n)

  ′ = b ⊲ f (z)(b ⊗ R n).

  ❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s ev(f ⊗ R N )(z ⊗ R n) = ev(f (z) ⊗ R n) = f (z)(1 B ⊗ R n) = e(1 B ⊲ z ⊗ R n)

  = e(z ⊗ R n). R Hom(N, P ) op ❆❣♦r❛✱ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r ♦ ❤♦♠✲❢✉♥t♦r ✐♥t❡r♥♦ ❡♠ R B Hom(B ⊗ R N, P ) e op M

  ✱ ❝❛♥♦♥✐❝❛♠❡♥t❡ ❝♦♠ ✳ ❆♥t❡s✱ ♥♦t❡ q✉❡ ❛ ❡s✲ op tr✉t✉r❛ ❞❡ R ✲❜✐♠ó❞✉❧♦ ❡♠ q✉❛❧q✉❡r B✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ N é ❞❛❞❛✱ op

  ′ ′ ′

  ∈ R = t(r)s(r ) ⊲ n ♣❛r❛ q✉❛✐sq✉❡r r, r ❡ n ∈ N✱ ♣♦r r · n · r ✳ P♦r✲ B Hom(B ⊗ R N, P ) −→ R Hom(N, P ) op op t❛♥t♦✱ ❞❡✜♥❛ ϕ : ✱ t❛❧ q✉❡ op

  ϕ(f )(n) = f (1 B ⊗ R n) B Hom(B ⊗ R N, P ) ✱ ♣❛r❛ q✉❛✐sq✉❡r f ∈ ❡ op n ∈ N

  ✳ ▼♦str❡♠♦s q✉❡ ϕ(f) é ♠♦r✜s♠♦ ❞❡ R ✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱ op

  ϕ(f )(r · n) = f (1 B ⊗ R r · n) op = f (1 B · r ⊗ R n) op = f (t(r) ⊗ R n) op = f (t(r) ⊲ (1 B ⊗ R n)) op = t(r) ⊲ f (1 B ⊗ R n) = t(r) ⊲ ϕ(f )(n), R N op

  ♣♦✐s ❛ ❡str✉t✉r❛ ❞❡ B✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ ❡♠ B ⊗ é ❞❛❞❛✱ ♣❛r❛

  ′ ′ ′ op op

  ∈ B ⊗ R n) = bb ⊗ R n q✉❛✐sq✉❡r b, b ❡ n ∈ N✱ ♣♦r b ⊲ (b ✳ ❙❡❣✉❡ q✉❡ op ϕ(f ) R Hom(N, P )

  ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ ❆❣♦r❛✱ ♣❛r❛ q✉❛✐sq✉❡r g ∈ ✱ b ∈ B ❡ n ∈ N✱ ❞❡✜♥❛

  −1

op op

  ϕ : R Hom(N, P ) −→ B Hom(B ⊗ R N, P ),

  −1 −1 op

  (g)(b ⊗ R n) = b ⊲ g(n) t❛❧ q✉❡ ϕ ✳ ▼♦str❡♠♦s q✉❡ ϕ ❡stá ❜❡♠ ❞❡✲ R Hom(N, P ) op ✜♥✐❞❛✳ P❛r❛ t❛♥t♦✱ ♣❛r❛ q✉❛✐sq✉❡r b ∈ B✱ n ∈ N ❡ g ∈

  −1 −1

  ϕ (g) : B × N −→ P ϕ (g)(b, n) = b ⊲ g(n) ✜①♦✱ ❞❡✜♥❛ ^ ✱ t❛❧ q✉❡ ^ ✳

  −1

  ϕ (g) ▼♦str❡♠♦s q✉❡ ^ é R✲❜❛❧❛♥❝❡❛❞❛✱ ❞❡ ❢❛t♦ t❡♠♦s

  ^

  −1

  ϕ (g)(b · r, n) = bt(r) ⊲ g(n) = b ⊲ (t(r) ⊲ g(n)) = b ⊲ g(t(r) ⊲ n) = b ⊲ g(r · n)

  ^

  −1 = ϕ (g)(b, r · n). −1

  (g) : B ⊗ R N −→ P P♦rt❛♥t♦✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ϕ ✱ q✉❡ s❛t✐s❢❛③

  −1 op

  ϕ (g)(b ⊗ R n) = b ⊲ g(n) ✳ ❆ss✐♠✱ t❡♠♦s

  −1 −1

  ϕ ◦ ϕ (g)(n) = ϕ(ϕ (g))(n)

  −1 op

  = ϕ (g)(1 B ⊗ R n) = 1 B ⊲ g(n) = g(n),

  ♣♦r ♦✉tr♦ ❧❛❞♦✱

  −1 −1 op op

  ϕ ◦ ϕ(f )(b ⊗ R n) = ϕ (ϕ(f ))(b ⊗ R n) = b ⊲ ϕ(f )(n) op = b ⊲ f (1 B ⊗ R n) op = f (b ⊗ R n). R e M

  ❯s❛♥❞♦ ❡ss❛ ✐❞❡♥t✐✜❝❛çã♦✱ ♦ ♠♦r✜s♠♦ ❛✈❛❧✐❛çã♦ ❡♠ ✱

  ′

op

  ev : B Hom(B ⊗ R N, P ) ⊗ R N −→ P, B Hom(B ⊗ R N, P ) op é ❞❛❞♦✱ ♣❛r❛ q✉❛✐sq✉❡r f ∈ ❡ n ∈ N✱ ♣♦r

  ′ op ev (f ⊗ R n) = f (1 B ⊗ R n). B −→ R M M e

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ ♣r❡s❡r✈❛ ❤♦♠✲❢✉♥t♦r❡s ✐♥t❡r♥♦s✳ P❛r❛ t❛♥t♦✱ ❝♦♥s✐❞❡r❡ ❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧ op P (f )(b⊗ R n) = f (b ⊗ R b ⊲n) ξ P : B Hom(B ⊗ R N, P ) −→ B Hom(B ⊗ R N, P ), op t❛❧ q✉❡ ξ (1) (2) ✳ P❡r❝❡❜❛ q✉❡ ♣❛r❛ q✉❛✐sq✉❡r f ∈ B Hom(B ⊗ R N, P )

  ✱ n ∈ N ❡ b ∈ B✱ t❡♠♦s op ξ P (f )(b ⊗ R n) = f (b ⊗ R b ⊲ n)

  (1) (2) op P (f ) = f ◦ Can N = f (Can N (b ⊗ R n)),

  ♦✉ s❡❥❛✱ ξ ✳ ❆❣♦r❛ ♥♦t❡ q✉❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❡q✉✐✲ B −→ R M M e ✈❛❧❡♥t❡ ❛♦ ❞❛ ❉❡✜♥✐çã♦ ✸✳✶✻ ♣❛r❛ ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ B Hom(B ⊗ R N, P ) B Hom(B ⊗ R op ❡ ♣❛r❛ ♦s ❤♦♠✲❢✉♥t♦r❡s ✐♥t❡r♥♦s ❡

  M M e N, p) B R

  ✱ ❡♠ ❡ ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ é ❝♦♠✉t❛t✐✈♦ ξ N P ⊗ R B Hom(B ⊗ R N, P ) ⊗ R N // P. B Hom(B ⊗ R N, P ) ⊗ R N B Hom(B ⊗ R N, P ) ⊗ R N Id ev // op ev

  B Hom(B ⊗ R N, P )

  ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r f ∈ ❡ n ∈ N✱ t❡♠♦s

  ′ ′

  ev (ξ P ⊗ R N )(f ⊗ R n) = ev (ξ P (f ) ⊗ R n) op

  = ξ P (f )(1 B ⊗ R n) = f (1 B ⊗ R n) P = ev(f ⊗ R n).

  ❙❡ ♠♦str❛r♠♦s q✉❡ ξ é ✉♠❛ ❜✐❥❡çã♦ ♣❛r❛ t♦❞♦ P s❡✱ ❡ s♦♠❡♥t❡ s❡✱ Can N N

  é ✉♠❛ ❜✐❥❡çã♦✱ ❝♦♥❝❧✉í♠♦s ❛ ❞❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱ s❡ Can

  −1 −1

  := − ◦ Can é ❜✐❥❡t✐✈❛✱ ❞❡✜♥❛ ξ P N ✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❛r❛ q✉❛✐sq✉❡r f ∈ B Hom(B ⊗ R N, P )

  ✱ t❡♠♦s

  −1 −1 −1

  ξ ◦ ξ P (f ) = ξ (f ◦ Can N ) = f ◦ Can N ◦ Can = f P P N B Hom(B ⊗ R N, P ) op ❡ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ t♦❞♦ g ∈ ✱ t❡♠♦s

  −1 −1 −1 ξ P ◦ ξ (g) = ξ P (g ◦ Can ) = g ◦ Can ◦ Can N = g. P P N N R N

  ❆❣♦r❛✱ s❡ ξ é ❜✐❥❡t✐✈❛✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✸✳✶✽ ❝♦♥s✐❞❡r❛♥❞♦ X = B⊗ ✱ op Y = B ⊗ R N X

  ❡ µ = ξ ✱ t❡♠♦s q✉❡ ξ X (I B N ) = I B N ◦ Can N = Can N

  ⊗ ⊗ R R N

  é ✉♠❛ ❜✐❥❡çã♦✳ P♦rt❛♥t♦✱ s❡ Can é ❜✐❥❡t✐✈❛✱ t❡♠♦s q✉❡ Can é ❜✐✲ X ❥❡t✐✈❛ ❡ ♣♦r s✉❛ ✈❡③ ξ é ✉♠❛ ❜✐❥❡çã♦✳ ▲♦❣♦✱ ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ B −→ R M M e

  ♣r❡s❡r✈❛ ❤♦♠✲❢✉♥t♦r❡s ✐♥t❡r♥♦s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ Can é ❜✐❥❡t✐✈❛✳ R

  ❆ ♥♦çã♦ ❞❡ ✉♠❛ × ✲❍♦♣❢ á❧❣❡❜r❛ é ♠❛✐s ❣❡r❛❧ q✉❡ ✉♠ ❍♦♣❢ ❛❧❣❡✲ ❜ró✐❞❡✱ ❝♦♥❢♦r♠❡ ♥♦ss♦ ♣ró①✐♠♦ ❡ ú❧t✐♠♦ t❡♦r❡♠❛✳ L , H R , S) ❚❡♦r❡♠❛ ✸✳✷✵ ❙❡❥❛ (H ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳ ❊♥tã♦ ❛ ❝❛♥ô✲ ♥✐❝❛ op op

  Can : H ⊗ L H −→ H ⊗ L

  H, h ⊗ L k 7−→ h ⊗ L h k

  (1) (2) L k 7−→ h ⊗ L S(h )k (1) (2) op

  é ❜✐❥❡t✐✈❛✱ ❝♦♠ ✐♥✈❡rs❛ h ⊗ ✳

  −1

  ❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ ♠♦str❡♠♦s ❛♥t❡s q✉❡ Can é ♠♦r✜s♠♦ ❞❡

  ′

  H ✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r h, h ❡ k ∈ H✱ t❡♠♦s

  −1 ′ −1 ′

  Can (h ⊲ (h ⊗ L k)) = Can (h h ⊗ L h k)

  (1) (2) (1) ′(1) (2) ′(2) op

  = h h ⊗ L S(h h )h k

  (1) (1) (2) (1) ′(1) ′(2) (2) op

  = h h ⊗ L S(h )S(h )h k

  (1) (1) (2) (1) ′(1) ′(2) (2) (2) op

  = h h ⊗ L S(h )S(h )h k

  (1) (2)

  = h

  (h ⊗ L k)) = Can(h

  (1) (2)

  ⊗ L h

  (1) (1)

  )k) = h

  (2)

  ⊗ L op S(h

  (1)

  −1

  (2)

  ♣♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s Can(Can

  (1 H ⊗ L k) = h ⊲ (1 H ⊗ L op k) = h ⊗ L op k,

  −1

  (h ⊲ (1 H ⊗ L k)) = h ⊲ Can

  −1

  k) = Can

  (2)

  S(h

  )k = h

  (1)

  (1)

  ) ⊗ L k = h ⊗ L k. ❙❛❜❡✲s❡ q✉❡ ♥❡♠ t♦❞❛ × R ✲❍♦♣❢ á❧❣❡❜r❛ ✈❡♠ ❞❡ ✉♠ ❜✐❛❧❣❡❜ró✐❞❡

  (2)

  · ε L (h

  (1)

  ) · k = h

  (2)

  ⊗ L ε L (h

  )k = h

  (1)

  (2)

  ⊗ L s L ◦ ε L (h

  (1)

  )k = h

  (2) (2)

  S(h

  (2) (1)

  ⊗ L h

  ⊗ L h

  (h

  (1)

  h

  (1)

  )k = h

  ′(2)

  ))h

  (2)

  ⊗ L op S(t R (ε R (h

  

′(1)

  (1)

  (2)

  ))k = h

  (2)

  )s R (ε R (h

  ′(2)

  ⊗ L op S(h

  

′(1)

  h

  s R (ε R (h

  ))h

  −1

  ′(2)

  (Can(h ⊗ L op k)) = Can

  −1

  ❉❡ss❛ ❢♦r♠❛✱ Can

  ′ ⊗ L k).

  (h

  −1

  )k) = h ⊲ Can

  ⊗ L op S(h

  ′(1)

  

′(1)

  )k = h ⊲ (h

  ′(2)

  ⊗ L op S(h

  ′(1)

  )k = hh

  ′(2)

  ⊗ L op S(h

  à ❡sq✉❡r❞❛ q✉❡ ❝♦♥st✐t✉✐ ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡✳ ❉❡ ❢❛t♦✱ ✉♠ ❝♦♥tr❛✲ ❡①❡♠♣❧♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✶✼❪✳

  ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s

  ❆♣❡s❛r ❞♦ ❡st✉❞♦ s♦❜r❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s ♥♦ s❡♥t✐❞♦ ❬✼❪ t❡r ✉♠ ♣❛s✲ s❛❞♦ ❜❛st❛♥t❡ ❝✉rt♦✱ ❤á ❞❡ ❝♦♥s✐❞❡r❛r q✉❡ ❤♦✉✈❡ ✉♠ ❜♦♠ ♣r♦❣r❡ss♦ ❞❡s❞❡ q✉❡ ❢♦✐ ✐♥✐❝✐❛❞♦✳ ❈✐t❛♠♦s ❝♦♠♦ ♣r♦❣r❡ss♦ ♦s s❡❣✉✐♥t❡s ❡st✉❞♦s✿ ❝♦♠ó❞✉❧♦s s♦❜r❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s ❡♠ ❬✼❪✱ t❡♦r✐❛ ❞❡ ✐♥t❡❣r❛✐s ❞❡ á❧❣❡✲ ❜r❛s ❞❡ ❍♦♣❢✱ q✉❡ ❢♦✐ ❣❡♥❡r❛❧✐③❛❞❛ ♣❛r❛ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s ❡♠ ❬✺❪✱ t❡♠♦s t❛♠❜é♠ ❡st✉❞♦s s♦❜r❡ t❡♦r✐❛ ❞❡ ●❛❧♦✐s ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s ❡♠ ❬✻❪✳ ❚❛✐s ❡st✉❞♦s ♥ã♦ ❢♦r❛♠ ✐♥❝❧✉í❞♦s ♥❡st❡ tr❛❜❛❧❤♦ q✉❡ t❡♠ ❝❛rát❡r ✐♥tr♦❞✉tó✲ r✐♦✱ ♠❛s ✜❝❛ ❝♦♠♦ ♣r♦♣♦st❛ ♣❛r❛ ❡st✉❞♦s ♣♦st❡r✐♦r❡s✳

  P♦r ♦✉tr♦ ❧❛❞♦✱ ❡①✐st❡♠ ❞✐✈❡rs♦s ❛s♣❡❝t♦s ❞❡ á❧❣❡❜r❛s ❞❡ ❍♦♣❢ q✉❡ ❛✐♥❞❛ ♥ã♦ ❢♦r❛♠ ✐♥✈❡st✐❣❛❞♦s ❝♦♠♦ ❡st❡♥❞❡r ♣❛r❛ ♦ â♠❜✐t♦ ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s✳ P♦r ❡①❡♠♣❧♦✱ ♥❛❞❛ ❛✐♥❞❛ ❢♦✐ ❢❡✐t♦ ♥♦ s❡♥t✐❞♦ ❞❡ ✉♠❛ t❡♦r✐❛ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❡ t❡♦r❡♠❛s ❡str✉t✉r❛✐s ❞❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s✳

  ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

  ❬✶❪ ▼✳▼✳❙✳ ❆❧✈❡s✱ ❊✳ ❇❛t✐st❛✿ ❆♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ ❍♦♣❢ ❛❧❣❡❜r❛s✿ ❆ ❝❛t❡❣♦r✐❝❛❧ ❛♣♣r♦❛❝❤✳ ❳❳■■■ ❇r❛③✐❧✐❛♥ ❆❧❣❡❜r❛ ▼❡❡t✐♥❣✳ ▼❛✲ r✐♥❣á✱ ❏✉❧② ✷✼t❤ t♦ ❆✉❣✉st ✶st✱ ✷✵✶✹✳

  ❬✷❪ ●✳ ❇¨o❤♠ ❛♥❞ ❑✳ ❙③❧❛❝❤á♥②✐✱ ❍♦♣❢ ❛❧❣❡❜r♦✐❞ s②♠♠❡tr② ♦❢ ❛❜s✲ tr❛❝t ❋r♦❜❡♥✐✉s ❡①t❡♥s✐♦♥s ♦❢ ❞❡♣t❤ ✷✱ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✸✷ ✭✶✶✮ ✭✷✵✵✹✮ ✹✹✸✸✲✹✹✻✹✳

  ❬✸❪ ●✳ ❇¨o❤♠ ❛♥❞ ❑✳ ❙③❧❛❝❤á♥②✐✱ ❍♦♣❢ ❛❧❣❡❜r♦✐❞s ✇✐t❤ ❜✐❥❡❝t✐✈❡ ❛♥t✐♣♦❞❡s✿ ❛①✐♦♠s✱ ✐♥t❡❣r❛❧s ❛♥❞ ❞✉❛❧s✱❏✳ ❆❧❣❡❜r❛ ✷✼✹ ✭✷✵✵✹✮✱ ✼✵✽✲✼✺✵✳

  ❬✹❪ ●✳ ❇¨o❤♠✱ ❆♥ ❛❧t❡r♥❛t✐✈❡ ♥♦t✐♦♥ ♦❢ ❍♦♣❢ ❛❧❣❡❜r♦✐❞✱ ✐♥✿ ❙✳ ❈❛❡♥❡♣❡❡❧✱ ❋✳ ❱❛♥ ❖②st❛❡②❡♥ ✭❊❞s✳✮✱ ❍♦♣❢ ❛❧❣❡❜r❛s ✐♥ ♥♦♥❝♦♠♠✉✲ t❛t✐✈❡ ❣❡♦♠❡tr② ❛♥❞ ♣❤②s✐❝s✳ ▲❡❝t✉r❡ ♥♦t❡s ✐♥ P✉r❡ ❆♣♣❧✳ ▼❛t❤✳ ❱♦❧✳ ✷✸✾✱ ❉❡❦❦❡r✱ ◆❡✇ ❨♦r❦✱ ✷✵✵✺✱ ♣♣✳ ✸✶✲✺✸✳

  ❬✺❪ ●✳ ❇¨o❤♠✱ ■♥t❡❣r❛❧ t❤❡♦r② ♦❢ ❍♦♣❢ ❛❧❣❡❜r♦✐❞s✱ ❆❧❣❡❜r✳ ❘❡♣r❡✲ s❡♥t✳ ❚❤❡♦r② ✽✭✹✮ ✭✷✵✵✺✮✱ ✺✻✸✲✺✾✾✳ ❬✻❪ ●✳ ❇¨o❤♠✱ ●❛❧♦✐s t❤❡♦r② ♦❢ ❍♦♣❢ ❛❧❣❡❜r♦✐❞s✱ ❆♥♥✳ ❯♥✐✈✳ ❋❡r✲ r❛r❛ ❙❡③✳ ❱■■ ✭◆❙✮ ✺✶ ✭✷✵✵✺✮✱ ✷✸✸✲✷✻✷✳ ❬✼❪ ●✳ ❇¨o❤♠✱ ❍♦♣❢ ❛❧❣❡❜r♦✐❞s✱ ■♥✿❍❛♥❞❜♦♦❦ ♦❢ ❛❧❣❡❜r❛✳ ❱♦❧✳ ✻✱

  ✶✼✸✳✷✸✺✱ ❊❧s❡✈✐❡r✴◆♦rt❤✲❍♦❧❧❛♥❞ ✭✷✵✵✾✮✳ A ❬✽❪ ❚✳ ❇r③❡③✐♥s❦✐ ❛♥❞ ●✳ ▼✐❧✐t❛r✉✱ ❇✐❛❧❣❡❜r♦✐❞s✱ × ✲❜✐❛❧❣❡❜r❛s

  ❛♥❞ ❞✉❛❧✐t②✱ ❏✳ ❆❧❣❡❜r❛ ✷✺✶ ✭✷✵✵✷✮✱ ✷✼✾✲✷✾✹✳ ❬✾❪ ❚✳ ❇r③❡③✐♥s❦✐ ❛♥❞ ❘✳ ❲✐s❜❛✉❡r✱ ❈♦r✐♥❣s ❛♥❞ ❝♦♠✉❞✉❧❡s✱ ▲♦♥✲

  ❞♦♥ ▼❛t❤✳ ❙♦❝✳ ▲❡❝t✉r❡ ◆♦t❡ ❙❡r✐❡s ✸✵✾✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈✳ Pr❡ss✳ ✭✷✵✵✸✮✳

  ❬✶✵❪ ❆✳ ❈♦♥♥❡s ❛♥❞ ❍✳ ▼♦s❝♦✈✐❝✐✱ ❘❛♥❦✐♥✲❈♦❤❡♥ ❜r❛❝❦❡ts ❛♥❞ t❤❡ ❍♦♣❢ ❛❧❣❡❜r❛ ♦❢ tr❛♥s✈❡rs❡ ❣❡♦♠❡tr②✱ ▼♦s❝✳ ▼❛t❤✳ ❏✳ ✹ ✭✷✵✵✹✮✱ ✶✶✶✲✶✸✵✳

  ❬✶✶❪ ❙✳ ❉˘as❝˘a❧❡s❝✉❀ ❈✳ ◆˘ast˘as❡s❝✉❀ ❙✳ ❘❛✐❛♥✳ ❍♦♣❢ ❆❧❣❡❜r❛s✿ ❆♥ ■♥tr♦❞✉❝t✐♦♥✱ ◆❡✇ ❨♦r❦✿ ▼❛r❝❡❧ ❉❡❦❦❡r✱ ✹✵✶♣✳ ✭✷✵✵✶✮✳

  ❬✶✷❪ P✳ ❍♦ ❍❛✐✳ ❚❛♥♥❛❦❛✲❑r❡✐♥ ❞✉❛❧✐t② ❢♦r ❍♦♣❢ ❛❧❣❡❜r♦✐❞s✱ ■sr✳ ❏✳ ▼❛t❤✳ ✶✻✼✭✶✮✱ ✶✾✸✲✷✷✺ ✭✷✵✵✽✮

  ❬✶✸❪ ❚✳ ❲✳ ❍✉♥❣❡r❢♦r❞✳ ❆❧❣❡❜r❛✱ ◆❡✇ ❨♦r❦✿ ❙♣r✐♥❣❡r✲ ❱❡r❧❛❣✱ ✺✵✷♣✳ ✭✷✵✵✵✮✳

  ❬✶✹❪ ▲✳ ❑❛❞✐s♦♥ ❛♥❞ ❑✳ ❙③❧❛❝❤á♥②✐✱ ❉✉❛❧s ❜✐❛❧❣❡❜r♦✐❞s ❢♦r ❞❡♣t❤ t✇♦ r✐♥❣ ❡①t❡♥s✐♦♥s✱ ♠❛t❤✳❘❆✴✵✶✵✽✵✻✼✳ ❬✶✺❪ ▲✳ ❑❛❞✐s♦♥ ❛♥❞ ❑✳ ❙③❧❛❝❤á♥②✐✱ ❇✐❛❧❣❡❜r♦✐❞ ❛❝t✐♦♥s ♦♥ ❞❡♣t❤ t✇♦ ❡①t❡♥s✐♦♥s ❛♥❞ ❞✉❛❧✐t②✱ ❆❞✈✳ ▼❛t❤✳ ✶✼✾ ✭✷✵✵✸✮✱ ✼✺✲✶✷✶✳ ❬✶✻❪ ❈✳ ❑❛ss❡❧✳ ◗✉❛♥t✉♠ ●r♦✉♣s✱ ◆❡✇ ❨♦r❦✿ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✺✸✶♣✳

  ✭✶✾✾✺✮✳ ❬✶✼❪ ❯✳ ❑rä❤♠❡r ❛♥❞ ❆✳ ❘♦✈✐✱ ❆ ▲✐❡✲❘✐♥❡❤❛rt ❛❧❣❡❜r❛ ✇✐t❤ ♥♦

  ❛♥t✐♣♦❞❡✱ ♣r❡♣r✐♥t ✭✷✵✶✸✮✱ ❛r❳✐✈✿ ✶✸✵✽✳✻✼✼✵✳ ❬✶✽❪ ❏✳❍✳ ▲✉✱ ❍♦♣❢ ❛❧❣❡❜r♦✐❞s ❛♥❞ q✉❛♥t✉♠ ❣r♦✉♣♦✐❞s✱ ■♥t❡r♥❛t✳

  ❏✳ ▼❛t❤✳ ✼ ✭✶✾✾✻✮✱ ✹✼✲✼✵✳ ❬✶✾❪ ❏✳ ▼✳ ▼♦♠❜❡❧❧✐✳ ❯♥❛ ✐♥tr♦❞✉❝✐ó♥ ❛ ❧❛s ❝❛t❡❣♦rí❛s t❡♥✲ s♦r✐❛❧❡s ② s✉s r❡♣r❡s❡♥t❛❝✐♦♥❡s✱ ◆♦t❛s ❞❡ ❛✉❧❛✱ ✽✵♣✳ ❉✐s✲

  ♣♦♥í✈❡❧ ❡♠ ❤tt♣✿✴✴✇✇✇✳❢❛♠❛❢✳✉♥❝✳❡❞✉✳❛r✴ ♠♦♠❜❡❧❧✐✴❝❛t❡❣♦r✐❛s✲ t❡♥s♦r✐❛❧❡s✸✳♣❞❢✳ ❬✷✵❪ ❏✳ ▼♦r❛✈❛✳ ◆♦❡t❤❡r✐❛♥ ❧♦❝❛❧✐s❛t✐♦♥s ♦❢ ❝❛t❡❣♦r✐❡s ♦❢ ❝♦❜♦r✲

  ❞✐s♠ ❝♦♠♦❞✉❧❡s✱ ❆♥♥✳ ♦❢ ▼❛t❤✳ ✭✷✮ ✶✷✶✭✶✾✽✺✮✱ ✶✲✸✾✳ ❬✷✶❪ ❏✳ ▼rˇc✉♥✳ ❚❤❡ ❍♦♣❢ ❛❧❣❡❜r♦✐❞s ♦❢ ❢✉♥❝t✐♦♥s ♦♥ ét❛❧❡ ❣r♦✉✲

  ♣♦✐❞s ❛♥❞ t❤❡✐r ♣r✐♥❝✐♣❛❧ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥❝❡✱ ❏✳ P✉r❡ ❆♣♣❧✳ ❆❧❣❡❜r❛ ✶✻✵ ✭✷✵✵✶✮✱ ✷✹✾✲✷✻✷✳

  ❬✷✷❪ ❉✳ ❘✳ P❛♥s❡r❛✳ ➪❧❣❡❜r❛s ❞❡ ❍♦♣❢ ❋r❛❝❛s✿ ❚❡♦r❡♠❛s ❞❡ ❉✉✲ ❛❧✐❞❛❞❡ ❡ ❞❡ ▼❛s❝❤❦❡✱ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦✱ ❯❋❙❈✳ ✷✵✶✸✳

  ❬✷✸❪ ❇✳ P❛r❡✐❣✐s✱ ❆ ♥♦❝♦♠♠✉t❛t✐✈❡ ♥♦♥❝♦❝♦♠♠✉t❛t✐✈❡ ❍♦♣❢ ❛❧✲ ❣❡❜r❛ ✐♥ ✧♥❛t✉r❡✧✱ ❏✳ ❆❧❣❡❜r❛ ✼✵ ✭✶✾✽✶✮✱ ✸✺✻✲✸✼✹✳

  ❬✷✹❪ ❉✳ ❘❛✈❡♥❡❧✳ ❈♦♠♣❧❡① ❝♦❜♦r❞✐s♠ ❛♥❞ st❛❜❧❡ ❤♦♠♦t♦♣② ❣r♦✉♣s ♦❢ s♣❤❡r❡s✱ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤✳ ❙❡r✐❡s✱ ✈♦❧✳ ✶✷✶✱ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ❙❛♥ ❉✐❡❣♦✱ ✶✾✽✻✳

  ❬✷✺❪ P✳ ❙❝❤❛✉❡♥❜✉r❣✱ ❉✉❛❧s ❛♥❞ ❞♦✉❜❧❡s ♦❢ q✉❛♥t✉♠ ❣r♦✉♣♦✐❞s R ✭× ✲❍♦♣❢ ❛❧❣❡❜r❛s✮✱ ❆▼❙ ❈♦♥t❡♠♣✳ ▼❛t❤✳ ✷✻✼ ♣♣ ✷✼✸✲✷✾✾✱ ❆▼❙✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✱ ✷✵✵✵✳

  ❬✷✻❪ P✳ ❙❝❤❛✉❡♥❜✉r❣✱ ❇✐❛❧❣❡❜r❛s ♦✈❡r ♥♦♥❝♦♠♠✉t❛t✐✈❡ r✐♥❣s ❛♥❞ ❛ str✉❝t✉r❡ t❤❡♦r❡♠ ❢♦r ❍♦♣❢ ❜✐♠♦❞✉❧❡s✱ ❆♣♣❧✳ ❈❛t❡❣✳ ❙tr✉✲ t✉r❡s ✻ ✭✶✾✾✽✮✱ ✶✾✸✲✷✷✷✳

  ❬✷✼❪ ❑✳ ❙③❧❛❝❤á♥②✐✱ ▼♦♥♦✐❞❛❧ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥❝❡✱ ❆▼❙ ❈♦♥t❡♠♣✳ ▼❛t❤✳ ✸✾✶ ♣♣ ✸✺✸✲✸✻✾✱ ❆▼❙✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✱ ✷✵✵✺✳

  ❬✷✽❪ ▼✳ ❚❛❦❡✉❝❤✐✱ ●r♦✉♣s ♦❢ ❛❧❣❡❜r❛s ♦✈❡r A ⊗ A✱ ❏✳ ▼❛t❤✳ ❙♦❝✳ ❏❛♣❛♥ ✷✾ ✭✶✾✼✼✮ ✹✺✾✲✹✾✷✳

  ❬✷✾❪ ▲✳ ❆✳ ❯❧✐❛♥❛✳ ❊q✉✐✈❛r✐❛♥t✐③❛çã♦ ❞❡ ❝❛t❡❣♦r✐❛s K✲❧✐♥❡❛r❡s✱ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦✱ ❯❋❙❈✳ ✷✵✶✺✳

  ❬✸✵❪ ❘✳ ❲✐s❜❛✉❡r✳ ❋♦✉♥❞❛t✐♦♥s ♦❢ ♠♦❞✉❧❡ ❛♥❞ r✐♥❣ t❤❡♦r②✱ ❆ ❍❛♥❞❜♦♦❦ ❢♦r ❙t✉❞② ❛♥❞ ❘❡s❡❛r❝❤✱ ❯♥✐✈❡rs✐t② ♦❢ ❉üss❡❧❞♦r❢✱ ✭✶✾✾✶✮✳

  ❬✸✶❪ P✳ ❳✉✱ ◗✉❛♥t✉♠ ❣r♦✉♣♦✐❞s✱ ❈♦♠♠✳ ▼❛t❤✳ P❤②s✳ ✷✶✻ ✭✷✵✵✶✮✱ ✺✸✾✲✺✽✶✳

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