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❛

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❯♥✐✈❡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

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❍♦♣❢ ❆❧❣❡❜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

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❆❣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❛❜❛❧❤♦✳

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

♥♦çã♦ ♠❛✐s ❣❡r❛❧ ❞♦ q✉❡ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡s✳

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❆❜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 t♦ ❜✐❛❧❣❡❜r♦✐❞✱ ❛s ❚❛❦❡✉❝❤✐✬s ×R✲❜✐❛❧❣❡❜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✐♥❣ ❢✉♥❝t♦rMB−→RMR ✐s str✐❝t❧② ♠♦♥♦✐❞❛❧ ❛♥❞ ❛♥❛❧♦❣♦✉s t♦ t❤❡ ❝❛s❡ ♦❢ ❝♦♠♦❞✉❧❡s ♦✈❡r ❜✐❛❧❣❡❜r♦✐❞s✳ ❆t t❤❡ ❡♥❞ ♦❢ t❤❡ ✇♦r❦ ✇❡ ♣r❡s❡♥t t❤❡ ♥♦t✐♦♥ ♦❢ ❛×R✲❍♦♣❢ ❛❧❣❡❜r❛✱ ♣r♦♣♦s❡❞ ❜② P✳ ❙❝❤❛✉❡♥❜✉r❣✱ ✇❤✐❝❤ ✐s ❛

♠♦r❡ ❣❡♥❡r❛❧ ❝♦♥❝❡♣t t❤❛♥ ♦❢ ❍♦♣❢ ❛❧❣❡❜r♦✐❞s✳

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(13)

❮♥❞✐❝❡

■♥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❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✾

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❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✶✼✸

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✶✼✹

(15)
(16)

■♥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⊗kA−→A⊗LA✱ ❡♠ q✉❡A❡Lsã♦ á❧❣❡❜r❛s s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦

❝♦♠ ✉♥✐❞❛❞❡k✳

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

(17)

❛❧❣❡❜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 H −→Hop

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 ❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧RMR ❞♦sR✲❜✐♠ó❞✉❧♦s✱ ❡♠ q✉❡Ré ✉♠❛ á❧❣❡❜r❛

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

❡R✲❝♦❛♥é✐s ✭♦❜❥❡t♦s ❝♦á❧❣❡❜r❛s ❡♠RMR✮✳

◆♦ ❝❛♣ít✉❧♦ ✷ ❡st✉❞❛♠♦s ❛ ♥♦çã♦ ❞❡ ❜✐❛❧❣❡❜ró✐❞❡s s❡❣✉♥❞♦ ❬✼❪✳ ❇✐✲ ❛❧❣❡❜ró✐❞❡✱ ✐♥✈❡♥t❛❞♦ ♣♦r ❚❛❦❡✉❝❤✐ ❝♦♠♦×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 ❞❡ ❛♥❡❧ ❡ ❝♦❛♥❡❧✱ ❛ á❧❣❡❜r❛ ❜❛s❡ ❞♦ ❛♥❡❧ ♥ã♦ éR✱ ♠❛sR⊗Rop✳ ❍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✐❞❡ é t❛♠❜é♠ ✉♠ ❜✐❛❧❣❡❜ró✐❞❡✳ ❆ s❛❜❡r✱ ♦ ❞✉❛❧ à ❡sq✉❡r❞❛∗B ❞❡ ✉♠ 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 ❞❡

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♦✉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 ✉♠❛ ❜✐á❧❣❡❜r❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ❝❛t❡❣♦r✐❛ MB ❞♦s B✲♠ó❞✉❧♦s à ❞✐✲ r❡✐t❛✱ é ♠♦♥♦✐❞❛❧ t❛❧ q✉❡ ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ MB −→ Mk 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 ❡sq✉❡❝✐♠❡♥t♦ ♣❛r❛ Mk é ❡str✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧✳ ◆♦ ✜♥❛❧ ❞♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❜✐❛❧❣❡❜ró✐❞❡s à ❡sq✉❡r❞❛ s❡❣✉♥❞♦ ●✳ ❇o❤♠ ❡♠ ❬✼❪✱ ❜✐❛❧❣❡❜ró✐❞❡s ❝♦♠ â♥❝♦r❛ ❡♠ ❬✸✶❪ ❡¨ ×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✐❞❡ H= (HL,HR, S)✱ ❛ tr✐♣❧❛((HR)op

cop,(HL)opcop, S)é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡

s♦❜r❡ á❧❣❡❜r❛s ❜❛s❡s Rop Lop✱ r❡s♣❡❝t✐✈❛♠❡♥t❡❀ ❙❡ ❛ ❛♥tí♣♦❞❛ S é

❜✐❥❡t✐✈❛✱ ❡♥tã♦((HR)op,(H

L)op, S−1)é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ s♦❜r❡ á❧❣❡✲

❜r❛s ❜❛s❡sR❡L✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ t❛♠❜é♠✱((HL)cop,(HR)cop, S−1) é ✉♠ ❍♦♣❢ ❛❧❣❡❜ró✐❞❡ s♦❜r❡ á❧❣❡❜r❛s ❜❛s❡sLop Rop✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❚❛♠❜é♠✱ ♠♦str❛♠♦s q✉❡ ❛ ❛♥tí♣♦❞❛ é ✉♠ ♠♦r✜s♠♦ ❞❡ ❜✐á❧❣❡❜ró✐❞❡s à ❡sq✉❡r❞❛ S : (HR)op

cop −→ HL. ◆♦ ✜♥❛❧ ❞♦ ❝❛♣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❛❜❛❧❤♦ s❡❣✉♥❞♦ ●✳ ❇¨o❤♠ ♣❡rt❡♥❝❡ ❛ ❝❧❛ss❡ ❞❡×R✲❍♦♣❢ 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✐♦✳

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❈❛♣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−→Cé ✉♠ ❢✉♥t♦r✱ ❝❤❛♠❛❞♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❡ ✶ é ✉♠ ♦❜❥❡t♦ ❡♠C✱ ❝❤❛♠❛❞♦ ♦❜❥❡t♦ ✉♥✐❞❛❞❡✳ ❆❧é♠ ❞✐ss♦✱aé ✉♠ ✐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❡ré ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ❡♥tr❡ ♦s ❢✉♥t♦r❡s − ⊗✶ ❡ Id ❞❡ C ♣❛r❛ C✱ t❛✐s q✉❡ ♣❛r❛ q✉❛✐sq✉❡r ♦❜❥❡t♦s

(20)

A, B, C ❡D∈C ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s

((A⊗B)⊗C)⊗D

aA⊗B,C,D

)

)

aABC⊗D

u

u

(A⊗(B⊗C))⊗D

aA,B⊗C,D

(A⊗B)⊗(C⊗D)

aABC⊗D

A⊗((B⊗C)⊗D)

A⊗aBCD

/

/A⊗(B⊗(C⊗D)),

(A⊗✶)⊗B aA,✶,B//

rA⊗B

'

'

A⊗(✶⊗B)

A⊗lB

A⊗B, sã♦ ❝♦♠✉t❛t✐✈♦s✳

❖ ❢❛t♦ ❞❡as❡r ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ♥♦s ❞✐③ q✉❡ ♣❛r❛ ❝❛❞❛ tr✐♣❧❛ ❞❡ ♦❜❥❡t♦sA, B, C ❡♠ C✱ t❡♠♦s ✉♠ ✐s♦♠♦r✜s♠♦

aA,B,C : (A⊗B)⊗C−→A⊗(B⊗C),

q✉❡ s❛t✐s❢❛③ ❞❡t❡r♠✐♥❛❞♦ ❞✐❛❣r❛♠❛✱ ✈❡r ✭❬✷✾❪✱ ♣á❣✳✷✹✮✳ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ t❡♠♦s ♦s ✐s♦♠♦r✜s♠♦slA:✶⊗A−→A❡rA:A⊗✶−→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, rsã♦ ❛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,,)♦✉ ❛♣❡♥❛sC✱ ❛♦ ✐♥✈és ❞❡(C,,✶, a, l, r)

(21)

❊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

aXY Z : (X×Y)×Z → X×(Y ×Z);

((x, y), z) 7→ (x,(y, z))

lX: {∗} ×X → X;

(∗, x) 7→ x

rX : X× {∗} → X.

(x,∗) 7→ x

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

❢❛t♦✱ ❝❧❛r♦ q✉❡a✱l❡rsã♦ ❜✐❥❡çõ❡s✱ ♦✉ s❡❥❛✱ ✐s♦♠♦r✜s♠♦s ❡♠Set✳ ▲♦❣♦✱ ❜❛st❛ ♠♦str❛r♠♦s ❛ ❝♦♥❞✐çã♦ ❞❛ ❞❡✜♥✐çã♦ ❞❡ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧✳ P❛r❛ t❛♥t♦✱ s❡❥❛♠A, B, C, A′, B, C❝♦♥❥✉♥t♦s q✉❛✐sq✉❡r ❡♠ Setf :

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×B)×C aA,B,C //

(f×g)×h

A×(B×C)

f×(g×h)

(A′×B)×C

aA′,B,C//′A

×(B×C),

{∗} ×A lA //

{∗}×f A f

{∗} ×A′

lA′

/

/A′

A× {∗} rA //

f×{∗} A f

A′× {∗}

rA′ //A

.

❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡ra∈A✱b∈B ❡c∈C✱ t❡♠♦s

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

= (f(a),(g×h)(b, c)) = (f(a),(g(b), h(c)))

=aA′,B,C′((f(a), g(b)), h(c))

=aA′,B,C′((f×g)(a, b), h(c))

=aA′,B,C′◦((f×g)×h)((a, b), c),

(22)

lA′(({∗} ×f)(∗, a)) =lA′(∗, f(a)) =f(a) =f(lA(∗, a)) =f ◦lA(∗, a) ❡

rA′((f× {∗})(a,∗)) =rA′(f(a),∗) =f(a) =f(rA(a,∗)) =f◦rA(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×aB,C,D)◦aA,B×C,D◦(aA,B,C×D)(((a, b), c), d)

= (A×aB,C,D)◦aA,B×C,D((a,(b, c)), d)

= (A×aB,C,D)(a,((b, c), d))

= (a,(b,(c, d))),

♣♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s

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

= (a,(b,(c, d))),

t❛♠❜é♠ t❡♠♦s

(A×lB)◦aA,{∗},B((a,∗), b) = (A×lB)(a,(∗, b))

= (a, lB(∗, b))

= (a, b) = (rA(a,∗), b)

= (rA×B)((a,∗), b).

P♦rt❛♥t♦✱(Set,×,{∗}, a, l, r)é ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳

❊①❡♠♣❧♦ ✶✳✹ ❆ ❝❛t❡❣♦r✐❛ V ectk ❞♦s ❡s♣❛ç♦s ✈❡t♦r✐❛s s♦❜r❡ ✉♠ ❝♦r♣♦

k é ♠♦♥♦✐❞❛❧✳ ❉❡✜♥✐♠♦s⊗:V ectk×V ectk−→V ectk ❝♦♠♦ ♦ ♣r♦❞✉t♦

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

V, U ❡W ∈V ectk ❡λ∈V✱ ❞❡✜♥✐♠♦s

aV U W : (V ⊗kU)⊗kW → V ⊗k(U⊗kW)

(v⊗u)⊗w 7→ v⊗(u⊗w);

lV : k⊗V → V

λ⊗v 7→ λv;

rV : V ⊗k → V

v⊗λ 7→ λv.

(23)

❊①❡♠♣❧♦ ✶✳✺ ❙❡❥❛ k ✉♠ ❝♦r♣♦✳ ❆ ❝❛t❡❣♦r✐❛ Supervectk ❞❡ s✉♣❡r✲ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s é ♠♦♥♦✐❞❛❧✳ ❖s ♦❜❥❡t♦s ❡♠Supervectk sã♦k✲❡s♣❛ç♦s ✈❡t♦r✐❛✐s✱ ❡q✉✐♣❛❞♦s ❞❡ ✉♠❛ ❣r❛❞✉❛çã♦ s♦❜r❡Z2✱ ♦✉ s❡❥❛✱ V =V0V1✳

❖s ♠♦r✜s♠♦s sã♦ tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s q✉❡ ♣r❡s❡r✈❛♠ ❛ ❣r❛❞✉❛çã♦✳ ❙❡V, W ∈ Supervectk ❞❡✜♥✐♠♦sV ⊗W :=V ⊗kW✱ ❝♦♠ ❣r❛❞✉❛çã♦

(V⊗W)0=V0⊗kW0⊕V1⊗kW1, (V⊗W)1=V0⊗kW1⊕V1⊗kW0.

❉❡✜♥✐♠♦s t❛♠❜é♠ a, l ❡ r ❝♦♠♦ ♥♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ❞❛ ❝❛t❡❣♦r✐❛ vectk✳

❊①❡♠♣❧♦ ✶✳✻ ❙❡❥❛♠k✉♠ ❝♦r♣♦ ❡R✉♠❛k✲á❧❣❡❜r❛✳ ❆ ❝❛t❡❣♦r✐❛RMR

❞♦s R✲❜✐♠ó❞✉❧♦s é ♠♦♥♦✐❞❛❧✳ ❉❡✜♥✐♠♦s

⊗:RMR×RMR−→RMR

❝♦♠♦ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❜❛❧❛♥❝❡❛❞♦ ♣♦rR✱ ❞❡♥♦t❛❞♦ ♣♦r⊗R✳ ❖ ♦❜❥❡t♦

✉♥✐❞❛❞❡ é ❛k✲á❧❣❡❜r❛ R✳ P❛r❛ q✉❛✐sq✉❡r M, N ❡ P ∈ RMR✱ m∈M✱

n∈N✱p∈P ❡ b∈R✱ ❞❡✜♥✐♠♦sa, l ❡ r✱ ♣♦r

aM N P : (M⊗RN)⊗RP → M⊗R(N⊗RP)

(m⊗n)⊗p 7→ m⊗(n⊗p);

lM : R⊗M → M

b⊗m 7→ b·m;

rM : M ⊗R → M

m⊗b 7→ m·b.

❊①❡♠♣❧♦ ✶✳✼ ❙❡❥❛ (H, µ, η,∆, ε) ✉♠❛ ❜✐á❧❣❡❜r❛ s♦❜r❡k ✉♠ ❝♦r♣♦✳ ❆ ❝❛t❡❣♦r✐❛HM ❞♦s H✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛ é ♠♦♥♦✐❞❛❧✳ ❉❡✜♥✐♠♦s⊗:H

M×HM−→HM ❝♦♠♦ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ s♦❜r❡ ♦ ❝♦r♣♦ ❞❡ ❜❛s❡k✳ ❙❡M, N ∈ HM❛ ❡str✉t✉r❛ ❞❡H✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ ❡♠M⊗N é ❞❛❞❛✱

♣❛r❛ t♦❞♦m∈M n∈N ❡h∈H✱ ♣♦r

h·(m⊗n) =h(1)·m⊗h(2)·n,

❡♠ q✉❡ ∆(h) = h(1) ⊗k h(2)✳ ❖s ♠♦r✜s♠♦s ♥❛t✉r❛✐s a, l, r sã♦ ♦s

♠❡s♠♦s ❞♦ ❡①❡♠♣❧♦ ❞❛ ❝❛t❡❣♦r✐❛V ectk✳ ❖ ♦❜❥❡t♦ ✉♥✐❞❛❞❡ é ♦ ❝♦r♣♦k

❝♦♠ ❛çã♦ ❞❡ H ❞❛❞❛ ♣❡❧❛ ❝♦✉♥✐❞❛❞❡✱ ♦✉ s❡❥❛✱ ♣❛r❛ q✉❛✐sq✉❡r h∈H ❡ λ∈kt❡♠♦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(1)·m)⊗(k(2)·n))

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= (h(1)·(k(1)·m))⊗(h(2)·(k(2)·n)) = (h(1)k(1)·m)⊗(h(2)k(2)·n) = ((hk)(1)·m)⊗((hk)(2)·n)

= (hk)·(m⊗n).

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

h·(k·λ) =h·(ε(k)λ) =ε(h)ε(k)λ

=ε(hk)λ

= (hk)·λ.

❱❛♠♦s ✈❡r✐✜❝❛r q✉❡aM N P : (M⊗N)⊗P −→M⊗(N⊗P)é ♠♦r✜s♠♦

❞❡ H✲♠ó❞✉❧♦s à ❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r M, N, P ∈ HM✱

m∈M✱n∈N✱p∈P ❡h∈H✱ t❡♠♦s

aM N P(h·((m⊗n)⊗p)) =aM N P(h(1)·(m⊗n)⊗h(2)·p)

=aM N P((h(1)(1)·m⊗h(1)(2)·n)⊗h(2)·p)

=aM N P((h(1)·m⊗h(2)(1)·n)⊗h(2)(2)·p)

=h(1)·m⊗(h(2)(1)·n⊗h(2)(2)·p) =h(1)·m⊗(h(2)·(n⊗p))

=h·(m⊗(n⊗p))

=h·aM N P((m⊗n)⊗p).

❚❛♠❜é♠ t❡♠♦s q✉❡ lM : k⊗M −→ M é ♠♦r✜s♠♦ ❞❡ H✲♠ó❞✉❧♦s à

❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡rM ∈ HM✱λ∈k❡h∈H✱ t❡♠♦s

lM(h·(λ⊗m)) =lM(h(1)·λ⊗h(2)·m)

=lM(ε(h(1))λ⊗h(2)·m)

=ε(h(1))λ(h(2)·m) =λε(h(1))h(2)·m

=λh·m

=h·(λm)

=h·lM(λ⊗m).

❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ♠♦str❛✲s❡ q✉❡rM :M⊗k−→M é ♠♦r✜s♠♦ ❞❡H✲

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

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❊①❡♠♣❧♦ ✶✳✽ ❙❡❥❛ H ✉♠❛ ❜✐á❧❣❡❜r❛ s♦❜r❡ k ✉♠ ❝♦r♣♦✳ ❆ ❝❛t❡❣♦r✐❛ MH ❞♦s H✲❝♦♠ó❞✉❧♦s à ❞✐r❡✐t❛ é ♠♦♥♦✐❞❛❧✳ ❉❡✜♥✐♠♦s

⊗:MH×MH −→MH

❝♦♠♦ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ s♦❜r❡ ♦ ❝♦r♣♦ ❞❡ ❜❛s❡⊗k✳ ❙❡M, N ∈MH✱ ❛

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

ρM N : M⊗N → (M ⊗N)⊗H

m⊗n 7→ m(0)n(0)m(1)n(1).

❖ ♦❜❥❡t♦ ✉♥✐❞❛❞❡ é ♦ ❝♦r♣♦ k❝♦♠ ❝♦❛çã♦ ❞❡H ❞❛❞❛✱ ♣❛r❛ t♦❞♦ λ∈k✱

♣♦r ρk : k −→ k⊗H✱ ρk(λ) = λ⊗1H✳ ❉❡✜♥✐♠♦s a, l, r ❝♦♠♦ ♥♦

❡①❡♠♣❧♦ ❞❛ ❝❛t❡❣♦r✐❛vectk

❱❛♠♦s ✈❡r q✉❡ aM N P : (M ⊗N)⊗P −→ M ⊗(N⊗P)é ♠♦r✜s♠♦

❞❡H✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❉❡ ❢❛t♦✱ s❡❥❛♠M, N ❡P ∈ MH q✉❛✐sq✉❡r✱

❡♥tã♦ ♣❛r❛ t♦❞♦m∈M✱n∈N ❡p∈P✱ q✉❡r❡♠♦s ♠♦str❛r q✉❡

(aM N P ⊗H)◦ρM⊗N,P =ρM,N⊗P◦aM N P.

❉❡ ❢❛t♦✱

(aM N P ⊗H)◦ρM⊗N,P((m⊗n)⊗p) =

= (aM N P ⊗H)((m⊗n)(0)⊗p(0)⊗(m⊗n)(1)p(1))

=m(0)⊗(n(0)⊗p(0))⊗m(1)n(1)p(1)

=m(0)⊗(n⊗p)(0)⊗m(1)(n⊗p)(1) =ρM,N⊗P(m⊗(n⊗p))

=ρM,N⊗P◦aM N P((m⊗n)⊗p).

❚❡♠♦s t❛♠❜é♠ q✉❡ lM : k⊗M −→ M ❡ rM : k⊗M −→ M sã♦

♠♦r✜s♠♦s ❞❡H✲❝♦♠ó❞✉❧♦ à ❞✐r❡✐t❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛❧q✉❡rM ∈MH ❡ ♣❛r❛ t♦❞♦m∈M ❡λ∈kt❡♠♦s

(lM ⊗H)◦ρk,M(λ⊗m) = (lM ⊗H)(λ⊗m(0)⊗1Hm(1))

=λm(0)⊗m(1)

=ρM(λm)

=ρM◦lM(λ⊗m).

(rM ⊗H)◦ρM,k(m⊗λ) = (rM ⊗H)(m(0)⊗λ⊗m(1)1H)

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=m(0)λ⊗m(1)

=ρM(mλ)

=ρM◦lM(m⊗λ).

◆ã♦ é ❞✐❢í❝✐❧ ♠♦str❛r q✉❡ a, l, rsã♦ ✐s♦♠♦r✜s♠♦s ♥❛t✉r❛✐s ❡ q✉❡ ✈❛❧❡♠ ♦s ❛①✐♦♠❛s ❞♦ ♣❡♥tá❣♦♥♦ ❡ ❞♦ tr✐â♥❣✉❧♦✳

❊①❡♠♣❧♦ ✶✳✾ ❈♦♥s✐❞❡r❡ C ✉♠❛ ❝❛t❡❣♦r✐❛✱ ❛ ❝❛t❡❣♦r✐❛ ❞❡ ❡♥❞♦❢✉♥t♦✲ r❡sEnd(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) α : F −→ G✱ β : F−→ Gtr❛♥s❢♦r♠❛çõ❡s

♥❛t✉r❛✐s✱ ❡♥tã♦ ♣❛r❛ t♦❞♦X ∈C✱ t❡♠♦s

(α⊗β) :F(F′(X))−→G(G′(X)),

t❛❧ q✉❡ (α⊗β)X = G(βX)◦αF′(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✐❛ ♠♦♥♦✐❞❛❧✳ ❆ ❝❛t❡❣♦r✐❛(Crev,rev,rev, arev, rrev, lrev)é ♠♦♥♦✐❞❛❧✱ ❡♠ q✉❡Crev=C

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

Y ⊗X✱ ♣❛r❛ t♦❞♦ X, Y ∈ C✳ ❊ t❛♠❜é♠ ♣❛r❛ q✉❛✐sq✉❡r X, Y, Z C t❡♠♦sarev

X,Y,Z =a

−1

Z,Y,X❀ rXrev=lX❀ lrevX =rX ❡ ✶rev=✶✳

✶✳✷ ❋✉♥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, ϕ0, φ) ❡♠ q✉❡ F : C −→ D é ✉♠ ❢✉♥t♦r ✭❝♦✈❛r✐❛♥t❡✮✱ ϕ0 : I −→ 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

I⊠F(A) lF(A) //

ϕ0⊠F(A)

F(A)

F(✶)⊠F(A)

φ✶A

/

/F(✶⊗A),

F(lA)

O

O F(A)⊠I

rF(A)

/

/

F(A)⊠φ0

F(A)

F(A)⊠F()

φA✶ //F(A⊗✶)

F(rA)

O

O

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F(A)⊠(F(B)F(C))F(A)⊠φB,C//F(A)⊠F(BC)

φA,B⊗C

(F(A)⊠F(B))F(C)

φAB⊠F(C)

aF(A)F(B)F(C)

O

O

F(A⊗(B⊗C))

F(A⊗B)⊠F(C)

φA⊗B,C

/

/F((A⊗B)⊗C).

F(aA,B,C)

O

O

❉❡✜♥✐çã♦ ✶✳✶✷ ❯♠ ❢✉♥t♦r ♠♦♥♦✐❞❛❧ (F, ϕ0, φ) é ❞✐t♦ s❡r ❢♦rt❡ ♦✉

❢♦rt❡♠❡♥t❡ ♠♦♥♦✐❞❛❧ s❡ϕ0é ✉♠ ✐s♦♠♦r✜s♠♦ ❡φé ✉♠ ✐s♦♠♦r✜s♠♦ ♥❛t✉r❛❧✳

❉❡✜♥✐çã♦ ✶✳✶✸ ❯♠ ❢✉♥t♦r ♠♦♥♦✐❞❛❧ (F, ϕ0, φ) é ❞✐t♦ s❡r ❡str✐t♦ ♦✉

❡str✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧ s❡ϕ0 ❡φ ❢♦r❡♠ ✐❞❡♥t✐❞❛❞❡✳

❊①❡♠♣❧♦ ✶✳✶✹ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ✜♥✐t♦✱ k ✉♠ ❝♦r♣♦ ❡ RepfG ❛ ❝❛✲

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♦ ❧✐♥❡❛rf :V Wt❛❧ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡rgGvV✱ t❡♠♦s

f(ρ(g)(v)) =σ(g)(f(v)).

❖ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦U :RepfGV ect

k é ❡str✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧✳

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

♣r❡s❡♥t❛çõ❡s(V, ρ)(W, σ) := (VW, ρσ)❡♠ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r

g∈G✱v⊗wt❡♠✲s❡ q✉❡(ρ⊗σ)(g)(v⊗w) =ρ(g)(v)⊗σ(g)(w)✱ é t❛♠❜é♠

✉♠❛ r❡♣r❡s❡♥t❛çã♦✳ ❚❡♠♦s t❛♠❜é♠ q✉❡ (k, ρk) é ✉♠❛ r❡♣r❡s❡♥t❛çã♦✱

❡♠ q✉❡ ρk(g)(1k) = 1k ♣❛r❛ t♦❞♦ g ∈ G✳ P♦rt❛♥t♦✱ k é ❛ ✉♥✐❞❛❞❡

♠♦♥♦✐❞❛❧✳ ❖s ♠♦r✜s♠♦s ❝❛♥ô♥✐❝♦saV,W,U(V⊗W)⊗U→V⊗(W⊗U)

lV : k⊗V→ V ❡ rV : V⊗k → V 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✉❡rv∈VwW uUgGrkt❡♠♦s

aV,W,U(((ρ⊗σ)⊗λ)(g)((v⊗w)⊗u))

=aV,W,U((ρ⊗σ)(g)(v⊗w)⊗λ(g)(u))

=aV,W,U((ρ(g)(v)⊗σ(g)(w))⊗λ(g)(u))

(28)

=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)(aV,W,U((v⊗w)⊗u)),

lV((ρk⊗ρ)(g)(r⊗v)) =lV(ρk(g)(r)⊗ρ(g)(v))

=lV(r⊗ρ(g)(v))

=rρ(g)(v) =ρ(g)(rv) =ρ(lV(r⊗v))

rV((ρ⊗ρk)(g)(v⊗r)) =rV(ρ(g)(v)⊗r)

=ρ(g)(v)r

=ρ(vr)

=ρ(rV(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♦ U :RepfGV ect

k é ❡str✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧✳

▼❛s ♥❡♠ s❡♠♣r❡ ♦ ❢✉♥t♦r ❡sq✉❡❝✐♠❡♥t♦ é ❡str✐t❛♠❡♥t❡ ♠♦♥♦✐❞❛❧✳ ❈♦♠♦ ✈❛♠♦s ✈❡r ♥♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦✳

❊①❡♠♣❧♦ ✶✳✶✺ ❙❡❥❛R✉♠❛ á❧❣❡❜r❛ s♦❜r❡ ♦ ❝♦r♣♦k❡U:RMR→V ectk

♦ ❢✉♥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♦✳

❉❡ ❢❛t♦✱ ♣❛r❛ q✉❛✐sq✉❡r M, N ∈ RMR✱ m ∈ M✱ n ∈ N ❡ λ ∈ k

❞❡✜♥✐♠♦s ❛ tr❛♥s❢♦r♠❛çã♦ ♥❛t✉r❛❧

φ:− ⊗k− ⇒ − ⊗R−,

t❛❧ q✉❡ φM,N(m⊗kn) =m⊗Rn❡ ♦ ♠♦r✜s♠♦sϕ0 é ❛ ✐♥❥❡çã♦

ϕ0: k → R. λ 7→ λ1R.

➱ 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♦

(29)

♥❛t✉r❛❧φ❡ ♦ ♠♦r✜s♠♦ϕ0♥ã♦ sã♦ ✐❞❡♥t✐❞❛❞❡s✳

k⊗kM

=

/

/

ϕ0⊗M

M

R⊗kM φR,M

/

/R⊗RM, lM

O

O M⊗kk

=

/

/

M⊗ϕ0

M

M ⊗kR φM,R

/

/M ⊗RR rM

O

O

M⊗k(N⊗kP) M⊗φN,P

/

/M ⊗k(N⊗RP) φM,N⊗R P

(M ⊗kN)⊗kP

φM,N⊗kP

aM,N,P

O

O

M ⊗R(N⊗RP)

(M ⊗RN)⊗kP

φM⊗R N,P

/

/(M⊗RN)⊗RP.

aM,N,P

O

O

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

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:AA−→A✱η:−→Asã♦ ♠♦r✜s♠♦s ❡♠C✱ s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ ✉♥✐t❛❧✐❞❛❞❡✱ ♦✉ s❡❥❛✱ ♦s s❡❣✉✐♥t❡s ❞✐✲ ❛❣r❛♠❛s ❝♦♠✉t❛♠

(A⊗A)⊗A µ⊗A//

aA,A,A

A⊗A

µ

A⊗(A⊗A)

A⊗µ

A⊗A µ //A

✭✶✳✶✮

(30)

✶⊗A η⊗A//

lA

$

$

A⊗A

µ

A⊗✶

A⊗η

o

o

rA

z

z

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❛❧

aA,A,A: (A⊗A)⊗A−→A⊗(A⊗A).

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

A⊗A⊗A µ⊗A//

A⊗µ

A⊗A

µ

A⊗A µ //A .

◆♦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✉✐ ❡❧❡♠❡♥t♦ ♥❡✉tr♦eA✳

❊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 ❡ η(∗) = eA✱ ♦s ❛①✐♦♠❛s ❞❡ ♦❜❥❡t♦ á❧❣❡❜r❛ s❡ tr❛❞✉③❡♠

❝♦♠♦

(ab)c=a(bc) ❡ eAa=a=aeA.

P♦rt❛♥t♦✱ A é ✉♠ ♠♦♥ó✐❞❡✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛❞♦ A ✉♠ ♠♦♥ó✐❞❡✱ ♦❜t❡♠♦s ✉♠ ♦❜❥❡t♦ á❧❣❡❜r❛ ❡♠ Set✳

❊①❡♠♣❧♦ ✶✳✶✽ ◆❛ ❝❛t❡❣♦r✐❛V ectk♦s ♦❜❥❡t♦s á❧❣❡❜r❛s sã♦ ❡①❛t❛♠❡♥t❡

❛sk✲á❧❣❡❜r❛s ✉♥✐t❛✐s✳

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

(31)

η:k−→A, η(1k) = 1A

k✲❜✐❧✐♥❡❛r❡s✱ ❝♦♠

(a·b)·c=a·(b·c) ❡ 1A·a=a=a·1A.

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

µ(a⊗b) =ab.

❖ ♠♦r✜s♠♦ ✉♥✐❞❛❞❡ é ❞❛❞♦ ♣♦r

η:k−→A, η(1k) = 1A.

➱ ❢á❝✐❧ ✈❡r✐✜❝❛r ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❞✐❛❣r❛♠❛s ✶✳✶ ❡ ✶✳✷✳

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

❉❡ ❢❛t♦✱ t❡♠♦s q✉❡l=r✱ ✈❡r ❬✶✻❪ ▲❡♠❛✳❳■✳✷✳✸✱ ❞❡✜♥❛µ:=l=r ❡η =Id✳ ❊♥tã♦ ♣❡❧♦ ❛①✐♦♠❛ ❞♦ tr✐â♥❣✉❧♦✱ ❛♣❧✐❝❛❞♦ ♣❛r❛ ❛ ✉♥✐❞❛❞❡ ♠♦♥♦✐❞❛❧ ✶✱ t❡♠♦s

µ(✶⊗µ) =µ(✶⊗l) =µ(r⊗✶) =µ(µ⊗✶), ❛ ✉♥✐t❛❧✐❞❛❞❡ é tr✐✈✐❛❧♠❡♥t❡ s❛t✐s❢❡✐t❛✳

❉❡✜♥✐çã♦ ✶✳✷✵ ❙❡❥❛♠ (A, µA, ηA) ❡ (B, µB, ηB) ♦❜❥❡t♦s á❧❣❡❜r❛ ❡♠

✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ C✳ ❯♠ ♠♦r✜s♠♦ ❞❡ ♦❜❥❡t♦s á❧❣❡❜r❛ f : A−→Bé ✉♠ ♠♦r✜s♠♦ ❡♠C✱ t❛❧ q✉❡ ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s ❝♦♠✉t❛♠✳

A⊗A f⊗f //

µA

B⊗B

µB

A

f //B

A f //B

✶.

ηA

O

O

ηB

>

>

❙❡❥❛k✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✉♥✐❞❛❞❡✱ ❡♥tã♦ ❛ ❝❛t❡❣♦r✐❛(kMk,⊗k, k)

é ✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧✳ ❊♠ q✉❡kMké ❛ ❝❛t❡❣♦r✐❛ ❞♦sk✲❜✐♠ó❞✉❧♦s✳

❯♠❛k✲á❧❣❡❜r❛ é ✉♠ ♦❜❥❡t♦ á❧❣❡❜r❛(R, µ, η)♥❛ ❝❛t❡❣♦r✐❛kMk✳

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

❞♦sR✲❜✐♠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✉❡Ré ✉♠❛ á❧❣❡❜r❛ s♦❜r❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦k✳ ❉❡✜♥✐çã♦ ✶✳✷✶ ❯♠R✲❛♥❡❧ é ✉♠ ♦❜❥❡t♦ á❧❣❡❜r❛(A, µ, η)♥❛ ❝❛t❡❣♦r✐❛

♠♦♥♦✐❞❛❧(RMR,⊗R, R)✳

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

❊♠ q✉❡ Aop é ♦ ♠❡s♠♦ k✲♠ó❞✉❧♦✱ Aop t❡♠ ❡str✉t✉r❛ ❞❡ Rop✲♠ó❞✉❧♦

à ❡sq✉❡r❞❛ ✭r❡s♣✳ à ❞✐r❡✐t❛✮ ✈✐❛ ❛ R✲❛çã♦ à ❞✐r❡✐t❛ ✭r❡s♣✳ à ❡sq✉❡r❞❛✮✱ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ é ❞❛❞❛ ♣♦rµop(a

Ropb) =µ(b⊗Ra)❡ ❛ ✉♥✐❞❛❞❡ 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 ❞✐✲ r❡✐t❛✱ é ✉♠ ♣❛r(M, θM)❡♠ q✉❡M é ✉♠ ♦❜❥❡t♦ ❡♠C❡θM:M⊗A→M

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

M ⊗A⊗A M⊗µ//

θM⊗A

M⊗A

θM

M⊗A

θM

/

/M,

M⊗✶

rM

%

%

M⊗η

/

/M ⊗A

θM

M.

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

❉❡✜♥✐çã♦ ✶✳✷✸ ❙❡❥❛♠(A, µ, η)✉♠ ♦❜❥❡t♦ á❧❣❡❜r❛ ❡♠C✉♠❛ ❝❛t❡❣♦r✐❛ ♠♦♥♦✐❞❛❧ ❡(M, θM)✱(N, θN)♠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✐✈♦✿

M ⊗A

θM

f⊗A

/

/N⊗A

θN

M

f //N.

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

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❊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✳

❉❡ ❢❛t♦✱ ❝♦♠❡ç❛♠♦s ❝♦♠ ✉♠A✲♠ó❞✉❧♦ à ❞✐r❡✐t❛(X, θX)❡ ❞❡✜♥✐♠♦s

φ : A −→ End(X)✱ ♣♦r φ(a)(x) = θX(x, a)✱ ♣❛r❛ q✉❛✐sq✉❡r x∈ X ❡

a, b∈A✳ ❊♥tã♦✱

φ(ab)(x) =θX(x, ab)

=θX(x, µ(a, b))

=θX◦(X, µ)(x, a, b)

=θX◦(θX, A)(x, a, b)

=θX(θX(x, a), b)

=φ(b)(φ(a)(x)).

P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛❞❛ ✉♠❛ ❛çã♦ à ❞✐r❡✐t❛ ❞❡ A ❡♠ X✱ ✈✐❛ φ : A −→

End(X)✱ ❡♥tã♦θX(x, a) :=φ(a)(x)❞❡✜♥❡ ✉♠❛ ❡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✉❡Cé ✉♠ ♦❜❥❡t♦ ❡♠ C ❡ ∆ : C −→ CC✱ ε : C −→ ✶ sã♦ ♠♦r✜s♠♦s ❡♠ C✱ s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ ❝♦✉♥✐t❛❧✐❞❛❞❡✱ ♦✉ s❡❥❛✱

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