❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❡ ❚❡❝♥♦❧♦❣✐❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
➪▲●❊❇❘❆❙ ❉❊ ❍❖P❋✱ ❖❇❏❊❚❖❙ ●❆▲❖■❙
❊ ■❉❊◆❚■❉❆❉❊❙ P❖▲■◆❖▼■❆■❙
❆❜❡❧ ●♦♠❡s ❞❡ ❖❧✐✈❡✐r❛ ❏ú♥✐♦r
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❲❛❧❞❡❝❦ ❙❝❤üt③❡r❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❡ ❚❡❝♥♦❧♦❣✐❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
➪▲●❊❇❘❆❙ ❉❊ ❍❖P❋✱ ❖❇❏❊❚❖❙ ●❆▲❖■❙
❊ ■❉❊◆❚■❉❆❉❊❙ P❖▲■◆❖▼■❆■❙
❆❜❡❧ ●♦♠❡s ❞❡ ❖❧✐✈❡✐r❛ ❏ú♥✐♦r
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❲❛❧❞❡❝❦ ❙❝❤üt③❡r❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
Gomes de Oliveira Júnior, Abel
Álgebras de Hopf, Objetos Galois e Identidades Polinomiais / Abel Gomes de Oliveira Júnior. -- 2018.
90 f. : 30 cm.
Dissertação (mestrado)-Universidade Federal de São Carlos, campus São Carlos, São Carlos
Orientador: Waldeck Shützer
Banca examinadora: Waldeck Shützer, Eliezer Batista, Humberto Luiz Talpo
Bibliografia
1. Álgebras de Hopf. 2. Objetos Galois. 3. Identidades Polinomiais. I. Orientador. II. Universidade Federal de São Carlos. III. Título.
Ficha catalográfica elaborada pelo Programa de Geração Automática da Secretaria Geral de Informática (SIn). DADOS FORNECIDOS PELO(A) AUTOR(A)
✐
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦
❛♦s ♠❡✉s ♣❛✐s✱ ❆❜❡❧ ❡ ❊♠í❧✐❛✱ ♣❡❧♦ ❛♣♦✐♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❡ ♣♦r t♦❞❛ ❛ ❝♦♥✜❛♥ç❛ ❡♠ t♦❞❛s ❛s ♠✐♥❤❛s ❞❡❝✐sõ❡s✱
❛♦s ♠❡✉s s❡t❡ ✐r♠ã♦s ❡ ✐r♠ãs ♣♦r t♦❞♦ ♦ ❝❛r✐♥❤♦ ❡ ❝✉✐❞❛❞♦ q✉❡ s❡♠♣r❡ t✐✈❡r❛♠ ❝♦♠ s❡✉ ✐r♠ã♦ ♠❛✐s ♥♦✈♦✱
à ♠✐♥❤❛ ❝♦♠♣❛♥❤❡✐r❛ ❇❡❛tr✐③✱ q✉❡ ❝♦♠♣❛rt✐❧❤❛ ♠❡✉s ❞✐❛s✱ ❛❧❡❣r✐❛s ❡ ♣r❡♦❝✉♣❛çõ❡s ❞❡s❞❡ ♦ ✐♥í❝✐♦ ❞♦ ♠❡str❛❞♦ ♣♦r t❛♥t♦s ♠♦♠❡♥t♦s ❞❡ ❢❡❧✐❝✐❞❛❞❡ ❡ ❡s♣❡❝✐❛❧♠❡♥t❡ ♣♦r ❞✐✈✐❞✐r ❝♦♠✐❣♦ ❛s ❤♦r❛s ❞❡ ❡st✉❞♦ ❧❛❞♦ ❛ ❧❛❞♦✳
❆❣r❛❞❡ç♦ ♦s ♠❡✉s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s✱ t❛♥t♦ ❛♦s ❞❡ ❧♦♥❣❛ ❞❛t❛ q✉❛♥t♦ ❛♦s ♠❛✐s r❡❝❡♥t❡s✱ ❝♦♠ ♦s q✉❛✐s t❡♥❤♦ ❝♦♠♣❛rt✐❧❤❛❞♦ ♥❡ss❡s ❛♥♦s t❛♥t♦s ♠♦♠❡♥t♦s ❞❡ ❛♣r❡♥❞✐③❛❣❡♠✱ ❛❝❛❞ê♠✐❝❛ ♦✉ ♥ã♦✳
❆♦s ♣r♦❢❡ss♦r❡s t❛♥t♦ ❞❛ ❣r❛❞✉❛çã♦ q✉❛♥t♦ ❞♦ ♠❡str❛❞♦ q✉❡ ♠❡ ❛❥✉❞❛♠ ❛ ❞❡s❝♦❜r✐r ❛ ❝❛❞❛ ❞✐❛ ♦ q✉❛♥t♦ ❛✐♥❞❛ t❡♥❤♦ q✉❡ ❛♣r❡♥❞❡r✳
❆❣r❛❞❡ç♦ t❛♠❜é♠✱ é ❝❧❛r♦✱ ❛♦ Pr♦❢✳ ❲❛❧❞❡❝❦ ❙❝❤üt③❡r ♣♦r t♦❞♦ ♦ s❡✉ t❡♠♣♦ ❞❡❞✐❝❛❞♦ ❛ ♠❡ ♦r✐❡♥t❛r t❛♥t♦ ♥♦ ✜♥❛❧ ❞❛ ❣r❛❞✉❛çã♦ q✉❛♥t♦ ♥❡st❡ ♠❡str❛❞♦ ❡ ❛ t♦❞♦s ♦s ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛ q✉❡ s❡ ❞✐s♣♦♥✐❜✐❧✐③❛r❛♠ ❛ ❧❡r ❡ ❝♦rr✐❣✐r ❡st❡ tr❛❜❛❧❤♦✳
✐✐
❘❡s✉♠♦
◆♦ss♦ ♦❜❥❡t✐✈♦ é ❡st✉❞❛r ❛s á❧❣❡❜r❛s ❞❡ ❍♦♣❢ ❡ ♦s ♦❜❥❡t♦s ●❛❧♦✐s s♦❜ ♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ q✉❛♥❞♦ ❡st❡s ♦❜❥❡t♦s sã♦ ❞✐st✐♥❣✉✐❞♦s ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦s✱ ♣♦r ♠❡✐♦ ❞❛s ❍✲✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s q✉❡ s❛t✐s❢❛③❡♠✳ P❛r❛ ✐ss♦ s❡❣✉✐♠♦s ♦ ♣r♦❣r❛♠❛ ✐♥✐❝✐❛❞♦ ♣♦r ❈✳ ❑❛ss❡❧ ❬✶✼❪✱ ♥♦ q✉❛❧ ❡❧❡ ❞❡✜♥❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❍✲✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ ✉♠ ❍✲❝♦♠ó❞✉❧♦ á❧❣❡❜r❛✱ ❡st❛❜❡❧❡❝❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s r❡s♣❡❝t✐✈♦s ❚✲✐❞❡❛✐s ❡ ✉t✐❧✐③❛ ❝❧❛ss✐✜❝❛çõ❡s ❞❡ ♦❜❥❡t♦s ●❛❧♦✐s✱ ❝♦♠♦ ❛ ❢❡✐t❛ ♣♦r ❇✐❝❤♦♥ ❬✺❪ ♣❛r❛ ❛s á❧❣❡❜r❛s ❞❡ ❍♦♣❢ ♠♦♥♦♠✐❛✐s ❞♦ t✐♣♦ ■✱ ♣❛r❛ ❡♥❝♦♥tr❛r ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s q✉❡ ❞✐st✐♥❣✉❡♠ t❛✐s ♦❜❥❡t♦s ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✳ ◆❡ss❡ ❝❛s♦✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❝✉❥♦ ♥ú❝❧❡♦ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ❚✲✐❞❡❛❧ ❞❛s ❍✲✐❞❡♥t✐❞❛❞❡s é ♦ ♣r✐♥❝✐♣❛❧ ❢❛t♦r ❞❡t❡r♠✐♥❛♥t❡✳
✐✐✐
❆❜str❛❝t
❖✉r ♠❛✐♥ ❣♦❛❧ ✐s t♦ st✉❞② ❍♦♣❢ ❛❧❣❡❜r❛s ❛♥❞ ●❛❧♦✐s ♦❜❥❡❝ts✱ ❢r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ✐❞❡♥t✐t✐❡s✱ ♠♦r❡ ♣r❡❝✐s❡❧② ❤♦✇ t❤❡s❡ ♦❜❥❡❝ts ❛r❡ ❞✐st✐♥❣✉✐s❤❡❞ ❜② t❤❡✐r ♣♦❧②♥♦♠✐❛❧ ❍✲✐❞❡♥t✐t✐❡s✳ ❚♦ t❤✐s ❡♥❞ ✇❡ ❢♦❧❧♦✇ t❤❡ ♣r♦❣r❛♠ st❛rt❡❞ ❜② ❈✳ ❑❛ss❡❧ ❬✶✼❪✳ ❚❤❡r❡ ❤❡ ❞❡✜♥❡s ♣♦❧②♥♦♠✐❛❧ ❍✲✐❞❡♥t✐t✐❡s t♦ ❍✲❝♦♠♦❞✉❧❡ ❛❧❣❡❜r❛s✱ ❣✐✈❡s s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡s❡ ❚✲✐❞❡❛❧s ❛♥❞ ✉s❡s ●❛❧♦✐s ♦❜❥❡❝ts ❝❧❛ss✐✜❝❛t✐♦♥s✱ ❛s ♠❛❞❡ ❜② ❇✐❝❤♦♥ ❬✺❪ ❢♦r t❤❡ ♠♦♥♦♠✐❛❧ ❍♦♣❢ ❛❧❣❡❜r❛s ♦❢ t②♣❡ ■✱ t♦ ✜♥❞ ♣♦❧②♥♦♠✐❛❧ ✐❞❡♥t✐t✐❡s t❤❛t ❞✐st✐♥❣✉✐s❤ t❤♦s❡ ♦❜❥❡❝ts✳ ■♥ t❤❛t ❝❛s❡✱ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ❛❧❣❡❜r❛ ❤♦♠♦♠♦r♣❤✐s♠ s✉❝❤ t❤❛t t❤❡ ❦❡r♥❡❧ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ❚✲✐❞❡❛❧ ♦❢ t❤❡ ❍✲✐❞❡♥t✐t✐❡s ✐s ❝r✉❝✐❛❧✳
✐✈
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ ❈♦♥❝❡✐t♦s Pr❡❧✐♠✐♥❛r❡s ✸
✷ ❈♦á❧❣❡❜r❛s ❡ ❈♦♠ó❞✉❧♦s ✾
✷✳✶ ➪❧❣❡❜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✷ ❈♦á❧❣❡❜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✸ ❉✉❛❧✐❞❛❞❡ ❡♥tr❡ ➪❧❣❡❜r❛s ❡ ❈♦á❧❣❡❜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✹ ▼ó❞✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✺ ❈♦♠ó❞✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✸ ❇✐á❧❣❡❜r❛s ❡ ➪❧❣❡❜r❛s ❞❡ ❍♦♣❢ ✷✸
✸✳✶ ❇✐á❧❣❡❜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✷ ➪❧❣❡❜r❛s ❞❡ ❍♦♣❢ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✹ ❍✲▼ó❞✉❧♦ ❡ ❍✲❈♦♠ó❞✉❧♦ ➪❧❣❡❜r❛s ✸✸
✹✳✶ ❆çõ❡s ❞❡ ➪❧❣❡❜r❛s ❞❡ ❍♦♣❢ ❡♠ ➪❧❣❡❜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✷ ❈♦❛çõ❡s ❞❡ ➪❧❣❡❜r❛s ❞❡ ❍♦♣❢ ❡♠ ➪❧❣❡❜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✺ ❖ Pr♦❞✉t♦ ❚♦r❝✐❞♦ ❡ ♦s ❖❜❥❡t♦s ●❛❧♦✐s ✹✷
✺✳✶ ❖ Pr♦❞✉t♦ ❚♦r❝✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✺✳✷ ❖s ❖❜❥❡t♦s H✲●❛❧♦✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✻ ❍✲■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ✺✸
✻✳✶ ■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✻✳✷ H✲■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
✻✳✸ ❉❡t❡❝t❛♥❞♦ H✲■❞❡♥t✐❞❛❞❡s P♦❧✐♥♦♠✐❛✐s ♣❛r❛ σH ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
✼ ❊①❡♠♣❧♦s ✻✻
✼✳✶ ❖s Hn2✲❝♦♠ó❞✉❧♦ á❧❣❡❜r❛s Aa,c ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
✼✳✷ ❆s ➪❧❣❡❜r❛s ❞❡ ❍♦♣❢ ▼♦♥♦♠✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✼✳✸ ➪❧❣❡❜r❛s ❞❡ ❍♦♣❢ ▼♦♥♦♠✐❛✐s ❞♦ ❚✐♣♦ ■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶
✶
■♥tr♦❞✉çã♦
❆s á❧❣❡❜r❛s ❞❡ ❍♦♣❢✱ ❛ss✐♠ ❝❤❛♠❛❞❛s ❡♠ ❤♦♥r❛ ❛ ❍❡✐♥③ ❍♦♣❢ ♣♦r s❡✉ tr❛❜❛❧❤♦ ♣✐♦♥❡✐r♦ ❬✶✹❪ ❡♠ ✶✾✹✶✱ ❢♦r❛♠ ✉s❛❞❛s ❝♦♠ ❡ss❡ ♥♦♠❡ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♣♦r ❆r♠❛♥❞ ❇♦r❡❧ ❬✼❪ ❡♠ ✶✾✺✸✳
◆♦ tr❛❜❛❧❤♦ ❞❡ ❍♦♣❢✱ sã♦ ✐♥tr♦❞✉③✐❞♦s ♦s✱ ♣♦st❡r✐♦r♠❡♥t❡ ❝❤❛♠❛❞♦s✱H✲❡s♣❛ç♦s✳ ❊st❡s
❝❛r❛❝t❡r✐③❛♠✲s❡ ♣♦r ♣♦ss✉ír❡♠ ✉♠❛ ♦♣❡r❛çã♦ ♣r♦❞✉t♦ ❡ ✉♠❛ ❡str✉t✉r❛ ❛❞✐❝✐♦♥❛❧ ❞❡H ❡♠ H⊗H✱ ❝♦♠ ❝♦♥❞✐çõ❡s ❞❡ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡✱ q✉❡ ❍♦♣❢ ♦❜s❡r✈♦✉ ✐♠♣♦r ❢♦rt❡s r❡str✐çõ❡s ♥❛
❡str✉t✉r❛ ❞❡H✱ ❛ ♣❛rt✐r ❞❛s q✉❛✐s ❡❧❡ ❞❡❞✉③✐✉ ✈ár✐♦s r❡s✉❧t❛❞♦s t♦♣♦❧ó❣✐❝♦s✳
❯♠❛ ✈✐sã♦ ♠❛✐s ❛❧❣é❜r✐❝❛ ❞❛s á❧❣❡❜r❛s ❞❡ ❍♦♣❢ s✉r❣✐✉ ❛❧❣✉♠ t❡♠♣♦ ❞❡♣♦✐s✱ ♥♦t❛✈❡❧♠❡♥t❡ ♥♦ ❧✐✈r♦ ❬✽❪ ❞❡ ❈❤❛s❡ ❡ ❙✇❡❡❞❧❡r ✭✶✾✻✾✮✳ ❏á s✉❛ ❡①♣❛♥sã♦ ❡ ♣♦♣✉❧❛r✐③❛çã♦ ♦❝♦rr❡✉ ♥❛s ❞é❝❛❞❛s s❡❣✉✐♥t❡s✱ ❝♦♠ ❛s ❛♣❧✐❝❛çõ❡s ❞❛ t❡♦r✐❛ ♦❜t✐❞❛s ♣♦r ❉r✐♥❢❡❧❞✳
❏á ❛ t❡♦r✐❛ ❞❡ ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ✭P■✮✱ ❛♣❡s❛r ❞❡ t❡r s✉❛ ❛❜♦r❞❛❣❡♠ ♠♦❞❡r♥❛ ❞❛❞❛ ♣♦r ❑❛♣❧❛♥s❦② ❡♠ ✶✾✹✽✱ t❡♠ s❡✉ ♣r✐♠❡✐r♦ ❛rt✐❣♦ ♣✉❜❧✐❝❛❞♦ ♣♦r ❉❡❤♥ ❡♠ ✶✾✷✷ ❡ ❞❡s❞❡ ❡♥tã♦ t❡♠ s✐❞♦ ❣❡♥❡r❛❧✐③❛❞❛ ❡ ❛t✉❛❧♠❡♥t❡ ❜♦❛ ♣❛rt❡ ❞❛ ✐♥✈❡st✐❣❛çã♦ s❡ ❞á s♦❜r❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ❝❡♥tr❛✐s✳ †
❯s❛♥❞♦ ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢ H✱ ♣♦❞❡✲s❡ ❞❡✜♥✐r ✉♠❛ H✲✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ✭❍P■✮
❡ é ❡ss❛ ✐♥t❡rs❡çã♦ q✉❡ ❞á ♦r✐❣❡♠ ❛ ❡ss❡ tr❛❜❛❧❤♦✳
❆❞♠✐t✐♠♦s ❞♦ ❧❡✐t♦r ❝♦♥❤❡❝✐♠❡♥t♦s s♦❜r❡ ❛s ❡str✉t✉r❛s ❛❧❣é❜r✐❝❛s ❜ás✐❝❛s ✭♠♦♥♦✐❞❡s✱ ❣r✉♣♦s✱ ❛♥é✐s✱ ❝♦r♣♦s✱ ♠ó❞✉❧♦s✱ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✮✱ ❞❡ ♠♦❞♦ q✉❡ ❛♣❡♥❛s ✉♠❛ ♦✉ ♦✉tr❛ ❞❡✜♥✐çã♦ ♦✉ r❡s✉❧t❛❞♦ ❡♥✈♦❧✈❡♥❞♦ ❡ss❡s ❝♦♥❝❡✐t♦s é ❝♦❧♦❝❛❞♦✱ ♠❛✐s ❝♦♠♦ ✉♠❛ r❡❝♦r❞❛çã♦ ❞♦ q✉❡ ❝♦♠♦ ✉♠❛ ♣r✐♠❡✐r❛ ✈✐sã♦ ❞♦ ❛ss✉♥t♦✳
❈♦♠❡ç❛♠♦s ❝♦♠ ✉♠ ❝❛♣ít✉❧♦ ❞❡ ❝♦♥❝❡✐t♦s ♣r❡❧✐♠✐♥❛r❡s q✉❡ ✈❡rs❛ ❜❛s✐❝❛♠❡♥t❡ s♦❜r❡ ♦❜❥❡t♦s ❞❛❞♦s ♣♦r ♣r♦♣r✐❡❞❛❞❡s ✉♥✐✈❡rs❛✐s✳
❖ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ tr❛③ ❡str✉t✉r❛s ❞✉❛✐s às á❧❣❡❜r❛s ❡ ♠ó❞✉❧♦s✿ ❝♦á❧❣❡❜r❛s ❡ ❝♦♠ó❞✉❧♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣❛r❛ q✉❡ ♥♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡ ♣♦ss❛♠♦s ✐♥tr♦❞✉③✐r ♦ ❝♦♥❝❡✐t♦ ❞❡ á❧❣❡❜r❛s ❞❡ ❍♦♣❢ ♣r♦♣r✐❛♠❡♥t❡✳
❖ ❝❛♣ít✉❧♦ q✉❛tr♦ tr❛③ ♦s ❝♦♥❝❡✐t♦s ❞❡ ❛çã♦ ❡ ❝♦❛çã♦ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ❍♦♣❢ ❡♠ ✉♠❛ á❧❣❡❜r❛ ❡ s❡rã♦ ❡str✉t✉r❛s ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♦s r❡s✉❧t❛❞♦s ✜♥❛✐s ❞♦ tr❛❜❛❧❤♦✳
❖ ♣ró①✐♠♦ ❝❛♣ít✉❧♦ tr❛③ ❞♦✐s ❝♦♥❝❡✐t♦s ❞✐st✐♥t♦s✳ ❖ ♣r✐♠❡✐r♦ é ♦ ♣r♦❞✉t♦ t♦r❝✐❞♦ q✉❡✱ ❞❛❞❛ ✉♠❛ á❧❣❡❜r❛✱ ❝r✐❛ ✉♠❛ ❡str✉t✉r❛ ❛❧❡r♥❛t✐✈❛ ❞❡ ♣r♦❞✉t♦✳ ❖ ♦✉tr♦ ❝♦♥❝❡✐t♦ sã♦ ♦s ♦❜❥❡t♦s ●❛❧♦✐s✳ ❊ss❡s t❛❧✈❡③ ♠❡r❡ç❛♠ ♠✉✐t♦ ♠❛✐s ❡s♣❛ç♦ ❞♦ q✉❡ r❡❝❡❜❡r❛♠✱ ❞❛❞❛ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ❡ ❡①t❡♥sã♦✱ ♠❛s ♣♦r q✉❡stã♦ ❞❡ ❢♦❝♦✱ ♣r♦❝✉r❛♠♦s ♥♦s ❛t❡r ❛♦ ❡ss❡♥❝✐❛❧✳ P♦r ❡①❡♠♣❧♦✱ ♦ ❧❡✐t♦r q✉❡ ❝♦♥❤❡ç❛ ❛s ✏❡①t❡♥sõ❡s ❢❡♥❞✐❞❛s✑ ✐rá ♥♦t❛r q✉❡ ❡❧❛s ♣❡r♠❡✐❛♠ t♦❞❛ ❛ s❡çã♦ ♠❛s s❡♠ ❛❜♦r❞á✲❧❛s ❞✐r❡t❛♠❡♥t❡✳
❖ s❡①t♦ ❝❛♣ít✉❧♦ é ♦ ♥ú❝❧❡♦ ❞♦ tr❛❜❛❧❤♦✳ ❈♦♠❡ç❛♠♦s ❝♦♠ ✉♠❛ ✐♥tr♦❞✉çã♦ ✐♥❣ê♥✉❛ à t❡♦r✐❛ ❞❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s s❡♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❞❛r ✉♠❛ ✈✐sã♦ ❞❡t❛❧❤❛❞❛ ❞♦ ❛ss✉♥t♦ ♠❛s tã♦ s♦♠❡♥t❡ ♠♦t✐✈❛r ❛ s❡çã♦ s❡❣✉✐♥t❡✳ ◆❡st❛ sã♦ ❞❡✜♥✐❞❛s ❛sH✲✐❞❡♥t✐❞❛❞❡s
†P❛r❛ ♠❛✐s s♦❜r❡ ❡ss❛ ♣❛rt❡ ❤✐stór✐❝❛ ✈❡❥❛ ♦s ❛rt✐❣♦s ✏P♦❧②♥♦♠✐❛❧ ■❞❡♥t✐t✐❡s✑ ❞♦ ❆♠✐ts✉r ❬✷❪ ❡ ✏❚❤❡
■♥tr♦❞✉çã♦ ✷
♣♦❧✐♥♦♠✐❛✐s ❡ ❛♣r❡s❡♥t❛❞❛s ❛❧❣✉♠❛s ❞❛s s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❆ ú❧t✐♠❛ s❡çã♦ ❡♥❝♦♥tr❛ ✉♠ ♠ét♦❞♦ ❜❛st❛♥t❡ út✐❧✱ ❛♣❡s❛r ❞❡ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ ❡♥❝♦♥tr❛r H✲✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡
✉♠❛ á❧❣❡❜r❛✳
❖ ú❧t✐♠♦ ❝❛♣ít✉❧♦ ❞❡st❡ tr❛❜❛❧❤♦ tr❛③ ❞♦✐s ❡①❡♠♣❧♦s ♥♦s q✉❛✐s ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦s r❡s✉❧t❛❞♦s ❞♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✳ ❖ ♣r✐♠❡✐r♦ ❞❡❧❡s t♦♠❛ H ❝♦♠♦ ❛ á❧❣❡❜r❛ ❞❡ ❚❛❢t ❡ ♦
✸
❈❛♣ít✉❧♦ ✶
❈♦♥❝❡✐t♦s Pr❡❧✐♠✐♥❛r❡s
❊♠ t♦❞♦ ❡st❡ t❡①t♦ k ✐♥❞✐❝❛ ✉♠ ❝♦r♣♦ ❛r❜✐trár✐♦✱ s❛❧✈♦ ♠❡♥çã♦ ❝♦♥trár✐❛✳ ❙❡♠♣r❡ q✉❡
s✉r❣✐r❡♠ k✲❡s♣❛ç♦s ✈❡t♦r✐❛✐s✱ ❛ ♠❡♥♦s q✉❡ s❡ ❞✐❣❛ ♦ ❝♦♥trár✐♦✱ ❡st❡s s❡rã♦ ❝❤❛♠❛❞♦s ❞❡
❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♦s t❡r♠♦s ❧✐♥❡❛r ❡ ❜✐❧✐♥❡❛r s❡rã♦ ❡♠♣r❡❣❛❞♦s ♥♦ ❧✉❣❛r ❞❡ k✲❧✐♥❡❛r ❡ k✲❜✐❧✐♥❡❛r✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ ✉♥✐❞❛❞❡† 1
k ❞♦ ❝♦r♣♦ k s❡rá s✐♠♣❧❡s♠❡♥t❡
✐♥❞✐❝❛❞❛ ♣♦r1✳
❆s ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s ♣❛r❛ ❡st❡ ❝❛♣ít✉❧♦ sã♦ ♦s ❧✐✈r♦s ❬✶✷❪✱ ❬✶✺❪ ❡ ❬✶✻❪✳
▲❡♠❜r❛r❡♠♦s ❜r❡✈❡♠❡♥t❡ ❛ s❡❣✉✐r ❞❡ ❛❧❣✉♥s ♦❜❥❡t♦s ❞❛❞♦s ♣♦r ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ q✉❡ s❡rã♦ ✐♠♣♦rt❛♥t❡s ♥❡st❡ t❡①t♦✳
❖ Pr♦❞✉t♦ ❚❡♥s♦r✐❛❧
❉❡✜♥✐çã♦ ✶✳✶ ✭Pr♦❞✉t♦ ❚❡♥s♦r✐❛❧✮✳ ❙❡❥❛♠V ❡W ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ ❉✐③❡♠♦s q✉❡(T, ϕ)✱
♥♦ q✉❛❧T é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡ϕ: V ×W −→T é ✉♠❛ ❛♣❧✐❝❛çã♦ ❜✐❧✐♥❡❛r✱ é ✉♠ ♣r♦❞✉t♦
t❡♥s♦r✐❛❧ ❡♥tr❡V ❡ W ✭s♦❜r❡ k✮ s❡ ♣❛r❛ t♦❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ Z ❡ t♦❞❛f: V ×W −→ Z
❜✐❧✐♥❡❛r✱ ❡①✐st❡ ú♥✐❝❛f¯:T −→Z ❧✐♥❡❛r t❛❧ q✉❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦✿
T
V ×W
Z
¯
f ϕ
f
▲❡♠❜r❡♠♦s q✉❡✱ ❝♦♠♦ t♦❞❛ ❞❡✜♥✐çã♦ ❞❛❞❛ ♣♦r ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧✱ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ é ú♥✐❝♦ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✱ ♠❛s é ♥❡❝❡ssár✐♦ ❝♦♥str✉✐r ✉♠ ♠♦❞❡❧♦ ❞❡ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧✳ ❊ss❛ ❝♦♥str✉çã♦ ✭❝❧áss✐❝❛✮ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❬✶✷✱ ❈❛♣ít✉❧♦ ✶✵❪ ♦✉ ❬✶✺✱ ❈❛♣ít✉❧♦ ■❱❪ ❡ r❡s✉♠✐❞❛♠❡♥t❡ ♣♦❞❡ s❡r ❢❡✐t❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❙❡❥❛ X ✉♠ ❝♦♥❥✉♥t♦✳ ❖ ❝♦♥❥✉♥t♦ kX = {f: X −→ k} é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠
♦♣❡r❛çõ❡s ❞❛❞❛s ♣♦r
• (f +g)(x) = f(x) +g(x) ♣❛r❛ t♦❞♦f, g ∈kX ❡x∈X✱
• (λf)(x) =λf(x)♣❛r❛ t♦❞♦ λ∈k✱ f ∈kX ❡x∈X✳
†❖ t❡r♠♦ ✏✉♥✐❞❛❞❡✑ é ❛♠❜í❣✉♦ ❡ ♣♦❞❡ ❢❛③❡r r❡❢❡rê♥❝✐❛ t❛♥t♦ ❛♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ q✉❛♥t♦
✶✳ ❈♦♥❝❡✐t♦s Pr❡❧✐♠✐♥❛r❡s ✹
❈♦♥s✐❞❡r❡ kX = {f: X −→ k | supp(f) é ✜♥✐t♦}✱ ❝♦♠ supp(f) = {x∈X|f(x)6= 0}✳ kX é ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ kX✳
❆❧é♠ ❞✐ss♦✱ ❞❛❞♦s V ❡ W ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✱ s❡ X = V ×W ✭♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❞❡
❝♦♥❥✉♥t♦s✮ ❡♥tã♦ kX =k(V ×W)é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡ t❡♠ ❝♦♠♦ ❜❛s❡
B =e(v,w), v ∈V, w ∈W ≡ {(v, w), v ∈V, w ∈W},
❝♦♠ ex: X−→k ❞❛❞❛ ♣♦r ex(y) =δxy†✱ ♣❛r❛ t♦❞♦ x, y ∈X✳
P♦r s✉❛ ✈❡③ ❝♦♥s✐❞❡r❡ U ♦ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ k(V ×W) ❣❡r❛❞♦ ♣❡❧❛ ✉♥✐ã♦ ❞♦s
s❡❣✉✐♥t❡s ❝♦♥❥✉♥t♦s✿
U1 ={(v1+v2, w)−(v1, w)−(v2, w), v1, v2 ∈V, w ∈W},
U2 ={(v, w1 +w2)−(v, w1)−(v, w2), v ∈V, w1, w2 ∈W},
U3 ={λ(v, w)−(λv, w), λ∈k, v, w∈W},
U4 ={λ(v, w)−(v, λw), λ∈k, v, w∈W}.
P♦❞❡♠♦s ❝♦♠ ✐ss♦ ❞❡✜♥✐r ✉♠ ❝♦♥❥✉♥t♦ ✭❞❡ ❝❧❛ss❡s ❧❛t❡r❛✐s✮
V ⊗kW ··=
k(V ×W)
U .
❊st❡ ♣♦ss✉✐ ✉♠❛ ú♥✐❝❛ ❡str✉t✉r❛ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ t❛❧ q✉❡ π: k(V ×W)−→V ⊗kW é
❧✐♥❡❛r✳††
➱ ❝♦rr✐q✉❡✐r♦ ♠♦str❛r q✉❡ V ⊗k W é ✉♠ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧✳ ❈♦♠♦ ❞❡ ♠♦❞♦ ❣❡r❛❧
✉s❛r❡♠♦s ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ s♦❜r❡ k✱ ♦♠✐t✐♠♦s ♦ í♥❞✐❝❡ k ❡ ❞❡♥♦t❛♠♦s s✐♠♣❧❡s♠❡♥t❡ V ⊗W✳ ❆ s❡❣✉✐r ❝♦❧♦❝❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s r❡❧❛❝✐♦♥❛❞♦s ❛ ❡ss❡ ❡s♣❛ç♦✿
❉❛❞♦sv ∈V ❡ w∈W ❞❡♥♦t❛r❡♠♦s ❛ ♣r♦❥❡çã♦ π(e(v,w))≡π(v, w) ✭❝♦♠ π ❛ ♣r♦❥❡çã♦
♥♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡✮ ♣♦rv⊗w✳ ❖s t❡♥s♦r❡s ❞❡ss❛ ❢♦r♠❛ sã♦ ❝❤❛♠❛❞♦s ❝❛♥ô♥✐❝♦s✱ ♣♦ré♠
♥♦t❡ q✉❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ V ⊗W ❡♠ ❣❡r❛❧ ♥ã♦ sã♦ ❝❛♥ô♥✐❝♦s ❡ s✐♠ ❝♦♠❜✐♥❛çõ❡s ❧✐♥❡❛r❡s
❞❡ t❡♥s♦r❡s ❝❛♥ô♥✐❝♦s✱ ♣♦r ❡①❡♠♣❧♦ ❞❛❞♦ {e1, e2} ❛ ❜❛s❡ ❝❛♥ô♥✐❝❛ ❞❡ V = R2✱ ❡♥tã♦
e1⊗e2+e2⊗e1 ∈R2⊗R2 ♥ã♦ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❞❛ ❢♦r♠❛ v⊗w♣❛r❛ ♥❡♥❤✉♠ v, w∈R2✳
✭❆♦ t❡♥t❛r ❢❛③ê✲❧♦ ❝❤❡❣❛♠♦s ❛ ✉♠ s✐st❡♠❛ ✐♠♣♦ssí✈❡❧✮✳
❉❛ ♣ró♣r✐❛ ❞❡✜♥✐çã♦ ❞♦ s✉❜❡s♣❛ç♦ U ❡ ❞♦ ❢❛t♦ ❞❡ π s❡r ❧✐♥❡❛r s❡❣✉❡♠ ❞❡ ✐♠❡❞✐❛t♦ ❛s
s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s ❞❡V ⊗W✿
✶✳ (v1+v2)⊗w=v1⊗w+v2⊗w✱ ♣❛r❛ t♦❞♦ v1, v2 ∈V ❡ w∈W✱
✷✳ v ⊗(w1+w2) = v⊗w1+v⊗w2✱ ♣❛r❛ t♦❞♦ v ∈V ❡ w1, w2 ∈W✱
✸✳ (λv)⊗w=λ(v⊗w) = v⊗(λw)✱ ♣❛r❛ t♦❞♦ v ∈V✱ w∈W ❡λ∈k✳
†❖ ❞❡❧t❛ ❞❡ ❑r♦♥❡❝❦❡r✿
δxy··= (
1, s❡y=x, 0, s❡y6=x.
♣❛r❛ t♦❞♦x, y∈X✳
††P❡❧❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✶✷✱ ✶✵✳✷ Pr♦♣✳ ✸❪ ✏❙❡❥❛♠V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱U ✉♠ s✉❜❡s♣❛ç♦
✶✳ ❈♦♥❝❡✐t♦s Pr❡❧✐♠✐♥❛r❡s ✺
Pr♦♣♦s✐çã♦ ✶✳✷✳ ❬✶✺✱ ❈♦r♦❧ár✐♦ ✺✳✸❪ ❙❡❥❛♠ V1✱ V2✱ W1 ❡ W2 ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ ❙❡
f1: V1 −→W1 ❡ f2: V2 −→ W2 sã♦ ❧✐♥❡❛r❡s ❡♥tã♦ ❡①✐st❡ ú♥✐❝❛ g: V1 ⊗V2 −→ W1 ⊗W2
❧✐♥❡❛r t❛❧ q✉❡g(v1⊗v2) = f1(v1)⊗f2(v2) ♣❛r❛ t♦❞♦ v1 ∈V1 ❡ v2 ∈V2✳
❉❡♥♦t❛r❡♠♦s ❛ ✭ú♥✐❝❛✮ ❛♣❧✐❝❛çã♦g ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ♣♦rf1⊗f2✳
▲❡♠❛ ✶✳✸✳ ❬✶✺✱ ❚❡♦r❡♠❛ ✺✳✶✶❪ ❙❡❥❛♠ V ❡ W ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❡ vi ∈ V✱ wi ∈ W ♣❛r❛
t♦❞♦ i= 1,2, . . . , n t❛✐s q✉❡
v1⊗w1+v2⊗w2+. . .+vn⊗wn = 0.
❙❡ w1, . . . , wn sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡♥tã♦ v1 =. . .=vn = 0✳
Pr♦♣♦s✐çã♦ ✶✳✹✳ ❬✶✺✱ ❈♦r♦❧ár✐♦ ✺✳✶✷❪ ❙❡❥❛♠V ❡W ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ ❙❡B ={vi, i∈I}
❡ C ={wj, j ∈J} sã♦ ❜❛s❡s ❞❡ V ❡ W✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡♥tã♦ {vi⊗wj, i∈I, j ∈J}
é ❜❛s❡ ❞❡ V ⊗W✳
❉❛❞♦sV✱V1✱V2 ❡W ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✱ ✐♥tr♦❞✉③✐♠♦s ❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s✿
Hom(V, W)··={α: V −→W, α ❧✐♥❡❛r}
Bil(V1, V2;W)··={β: V1×V2 −→W, β ❜✐❧✐♥❡❛r}
Mult(V, W, n)··={β: V| ×. . .{z ×V}
n✲✈❡③❡s
−→W, β n✲❧✐♥❡❛r}
❆ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ r❡❧❛❝✐♦♥❛ ♦s ♦♣❡r❛❞♦r❡s Bil ❡Hom✿
Pr♦♣♦s✐çã♦ ✶✳✺✳ ❬✶✷✱ ✶✵✳✹ ❚❡♦r❡♠❛ ✶✵❪ ❙❡❥❛♠V1✱ V2 ❡W ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❡(V1⊗V2, ϕ)
♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❡♥tr❡ V1 ❡ V2✳ ❊♥tã♦
Bil(V1, V2;W)= Hom(∼ V1⊗V2, W).
Pr♦♣♦s✐çã♦ ✶✳✻✳ ❬✶✷✱ ✶✵✳✹ Pr♦♣♦s✐çã♦ ✷✵❪ ❙❡ (V ⊗W, ϕ1) ❡ (W ⊗V, ϕ2) sã♦ ♣r♦❞✉t♦s
t❡♥s♦r✐❛✐s ❡♥tr❡ ♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s V ❡ W ❡♥tã♦ V ⊗W ∼=W ⊗V.
❆ ♣❛rt✐r ❞❛q✉✐✱ ❡♠ t♦❞♦ ♦ t❡①t♦✱ τ ✐rá ❞❡♥♦t❛r ♦ ✐s♦♠♦r✜s♠♦ ❞❛ ♣♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ ❛
s❛❜❡r✱ τ: V ⊗W −→W ⊗V ❞❛❞♦ ♣♦r τ(v⊗w) = w⊗v✱ ♣❛r❛ t♦❞♦ v ∈V ❡ w∈W✳
P♦❞❡♠♦s ❡st❡♥❞❡r ✐♥❞✉t✐✈❛♠❡♥t❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❡♥tr❡ ✈ár✐♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ ◆♦ ❝❛s♦n= 3✱ ❞❛❞♦s V1✱ V2 ❡ V3 ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ♦❜t❡♠♦s((V1⊗V2)⊗V3, ϕ1)
❡ (V1⊗(V2 ⊗V3), ϕ2)✱ ♣♦ré♠ ❡st❡s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s sã♦ ❝❛♥♦♥✐❝❛♠❡♥t❡ ✐s♦♠♦r❢♦s✱ ❝♦♠♦
❛✜r♠❛ ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✿
Pr♦♣♦s✐çã♦ ✶✳✼✳ ❬✶✷✱ ✶✵✳✹ ❈♦r♦❧ár✐♦ ✶✺❪ ❙❡❥❛♠V1✱ V2 ❡ V3 ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ ❙❡ ((V1⊗
V2)⊗V3, ϕ1) ❡ (V1⊗(V2⊗V3), ϕ2) sã♦ ♣r♦❞✉t♦s t❡♥s♦r✐❛✐s ❡♥tã♦
(V1⊗V2)⊗V3 =∼V1 ⊗(V2⊗V3).
❊♠ ✈✐rt✉❞❡ ❞❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡①♣r❡ss❛ ♣♦r ❡ss❛ ♣r♦♣♦s✐çã♦ ❢❛③ s❡♥t✐❞♦ ✐♥tr♦❞✉③✐r ❛ s❡❣✉✐♥t❡ ♥♦t❛çã♦ ♣❛r❛ ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❛ss♦❝✐❛❞♦ ❛♦ t❡♥s♦r ❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧V n ✈❡③❡s
✭n ≥0✮✿
V⊗n =
k, s❡ n= 0,
V ⊗. . .⊗V
| {z }
n✲✈❡③❡s
✶✳ ❈♦♥❝❡✐t♦s Pr❡❧✐♠✐♥❛r❡s ✻
Pr♦♣♦s✐çã♦ ✶✳✽✳ ❬✶✷✱ ✶✵✳✹ ❊①❡♠♣❧♦ ✼❪ ❙❡❥❛V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❙❡(k⊗V, ϕ)é ♣r♦❞✉t♦
t❡♥s♦r✐❛❧ ❡♥tã♦
k⊗V ∼=V ∼=V ⊗k.
▲❡♠❜r❛r q✉❡ ❞❛❞♦s ❞♦✐s ❡s♣❛ç♦s ✈❡t♦r✐❛✐sW1❡W2 ❛ s♦♠❛ ❞✐r❡t❛W1⊕W2 é ✉♠ ❡s♣❛ç♦
✈❡t♦r✐❛❧✱ ♠♦t✐✈❛ ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✿
Pr♦♣♦s✐çã♦ ✶✳✾✳ ❬✶✷✱ ✶✵✳✹ ❚❡♦r❡♠❛ ✶✼❪ ❙❡❥❛♠ V✱ W1 ❡ W2 ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ ❙❡ ((V ⊗
(W1⊕W2)), ϕ1) ❡ ((V ⊗W1)⊕(V ⊗W2), ϕ2) sã♦ ♣r♦❞✉t♦s t❡♥s♦r✐❛✐s ❡♥tã♦
V ⊗(W1⊕W2)∼= (V ⊗W1)⊕(V ⊗W2).
❉❡✜♥✐çã♦ ✶✳✶✵ ✭k✲á❧❣❡❜r❛✮✳ ❯♠❛ k✲á❧❣❡❜r❛ A é ✉♠ ❛♥❡❧ A ❝♦♠ ✉♥✐❞❛❞❡ ❥✉♥t♦ ❝♦♠ ✉♠
❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥é✐s φ: k −→A q✉❡ ❧❡✈❛1 ❡♠ 1A t❛❧ q✉❡ Im(φ)⊆Z(A)✱ ♦ ❝❡♥tr♦ ❞❡
A✳
Pr♦♣♦s✐çã♦ ✶✳✶✶✳ ❬✶✷✱ Pr♦♣♦s✐çã♦ ✶✵✳✷✶❪ ❉❛❞❛s k✲á❧❣❡❜r❛s A ❡ B✱ A ⊗ B ❝♦♠
♠✉❧t✐♣❧✐❝❛çã♦ ❞❛❞❛ ♣♦r
f: (A⊗B)×(A⊗B) −→ A⊗B
(a⊗b, a′⊗b′) 7−→ aa′⊗bb′
❡ ✉♥✐❞❛❞❡ 1A⊗1B é ✉♠❛ á❧❣❡❜r❛✱ ❝❤❛♠❛❞❛ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❞❛s á❧❣❡❜r❛sA ❡ B✳
❆ ➪❧❣❡❜r❛ ❆ss♦❝✐❛t✐✈❛ ▲✐✈r❡
❆ ❞❡✜♥✐çã♦ ❞❡ ♦❜❥❡t♦ ❧✐✈r❡ ❛♣❧✐❝❛❞♦ à ❝❛t❡❣♦r✐❛ ❞❛s á❧❣❡❜r❛s ♥♦s ❞❛ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿ ❉❡✜♥✐çã♦ ✶✳✶✷ ✭➪❧❣❡❜r❛ ❆ss♦❝✐❛t✐✈❛ ▲✐✈r❡✮✳ ❙❡❥❛♠ A ✉♠❛ á❧❣❡❜r❛✱ X ✉♠ ❝♦♥❥✉♥t♦ ❡ i:X −→A ✉♠❛ ❢✉♥çã♦✳ ❉✐③❡♠♦s q✉❡ A é ✉♠❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❧✐✈r❡ ❡♠ X s❡✱ ♣❛r❛
t♦❞❛ á❧❣❡❜r❛B ❡f: X −→B ❢✉♥çã♦✱ ❡①✐st❡ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛sf¯: A−→B
t❛❧ q✉❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦
A
X
B
¯
f i
f
❙❡♥❞♦ ❞❛❞❛ ♣♦r ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧✱ ✉♠❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❧✐✈r❡ ❡♠ X é ú♥✐❝❛
❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✳ ❯♠ ♠♦❞❡❧♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✶✻✱ ❈❛♣ít✉❧♦ ■❪ ❝♦♠♦ ❛ s❡❣✉✐r✿
❙❡❥❛ X ✉♠ ❝♦♥❥✉♥t♦✳ ❈♦♥s✐❞❡r❡ ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ khXi ❝♦♠ ❜❛s❡ ❞❛❞❛ ♣♦r t♦❞❛s
❛s ♣❛❧❛✈r❛s xi1, . . . , xip ♥♦ ❛❧❢❛❜❡t♦ X✱ ✐♥❝❧✉✐♥❞♦ ❛ ♣❛❧❛✈r❛ ✈❛③✐❛ ∅✳ ❆ ❝♦♥❝❛t❡♥❛çã♦ ❞❡
♣❛❧❛✈r❛s ❞❡✜♥❡ ✉♠❛ ♠✉❧t✐♣❧✐❝❛çã♦ ✭❛ss♦❝✐❛t✐✈❛✮ ❡♠khXi ❞❛❞❛ ♣♦r
(xi1, . . . , xip)(xip+1, . . . , xin) =xi1, . . . , xipxip+1, . . . , xin
❡ ❛ ✉♥✐❞❛❞❡ é ❞❛❞❛ ♣❡❧❛ ♣❛❧❛✈r❛ ✈❛③✐❛ 1 = ∅✳ ❈♦♥s✐❞❡r❡ A = khXi ❡ i: X −→ khXi ❛
✐♥❝❧✉sã♦✳ (khXi, i) é ✉♠❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❧✐✈r❡ ❡♠ X✳
◆♦t❡ q✉❡ khXi ♥❛❞❛ ♠❛✐s é q✉❡ ❞♦ q✉❡ ❛ á❧❣❡❜r❛ ❞♦s ♣♦❧✐♥ô♠✐♦s ❡♠ ✈❛r✐á✈❡✐s ♥ã♦
❝♦♠✉t❛t✐✈❛s ❞❡X✳ ❆ss✐♠✱ ♣♦r ❡①❡♠♣❧♦✱ s❡ X ={x1, x2}✱ x1x2−x2x1 é ✉♠ ❡❧❡♠❡♥t♦ ❞❡
✶✳ ❈♦♥❝❡✐t♦s Pr❡❧✐♠✐♥❛r❡s ✼
❆ ➪❧❣❡❜r❛ ❚❡♥s♦r✐❛❧
❉❡✜♥✐çã♦ ✶✳✶✸ ✭➪❧❣❡❜r❛ ❚❡♥s♦r✐❛❧✮✳ ❙❡❥❛V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❉✐③❡♠♦s q✉❡ (T(V), i)✱
♥♦ q✉❛❧ T(V) é ✉♠❛ á❧❣❡❜r❛ ❡ i: V −→T(V) ❧✐♥❡❛r✱ é ✉♠❛ á❧❣❡❜r❛ t❡♥s♦r✐❛❧ ❞❡ V s❡
♣❛r❛ t♦❞❛ á❧❣❡❜r❛ A ❡ t♦❞❛ ❛♣❧✐❝❛çã♦ f: V −→ A ❧✐♥❡❛r✱ ❡①✐st❡ ú♥✐❝♦ f¯: T(V) −→ A
❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s t❛❧ q✉❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦✿
T(V)
V
A
¯
f i
f
❙❡♥❞♦ ❞❛❞❛ ♣♦r ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧✱ ✉♠❛ á❧❣❡❜r❛ t❡♥s♦r✐❛❧ ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧V
é ú♥✐❝❛ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✳ ❯♠ ♠♦❞❡❧♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✶✻✱ ❈❛♣ít✉❧♦ ■■❪ ❝♦♠♦ ❛ s❡❣✉✐r✿
❙❡❥❛V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❉❡✜♥❛
T0(V) =k, T1(V) =V,
✳✳✳ ✳✳✳
Tn(V) =V⊗n.
❉❡♥♦t❡
T(V) =M
n≥0
Tn(V)
❡ ❞❡✜♥❛i:V −→T(V)♣♦ri(v) =v ∈T1(V)♣❛r❛ t♦❞♦v ∈V✳ ❉❛❞♦sx=v
1⊗. . .⊗vn ∈
Tn(V)❡ y=w
1 ⊗. . .⊗wr∈Tr(V)❞❡✜♥❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠ T(V)♣♦r
x·y=v1 ⊗. . .⊗vn⊗w1 ⊗. . .⊗wr∈Tn+r(V)
❡ ❡st❡♥❞❡♥❞♦ ♣♦r ❧✐♥❡❛r✐❞❛❞❡✳ T(V) é ✉♠❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❡ ✉♥✐tár✐❛ ❝♦♠ ❡ss❛
♠✉❧t✐♣❧✐❝❛ç❛♦ ❡ t♦♠❛♥❞♦ 1k ∈ T0(V) ❝♦♠♦ ✉♥✐❞❛❞❡✳ (T(V), i) é ❛ á❧❣❡❜r❛ t❡♥s♦r✐❛❧
❞❡ V✳ ➱ ❝♦♠✉♠ ❞❡♥♦t❛r x · y ♣♦r x ⊗ y✱ ❥á q✉❡ ❡①✐st❡ ✉♠ ✐s♦♠♦r✜s♠♦ ❝❛♥ô♥✐❝♦ Tn(V)⊗Tr(V)∼=Tn+r(V)✳
Pr♦♣♦s✐çã♦ ✶✳✶✹✳ ❬✶✻✱ Pr♦♣♦s✐çã♦ ■■✳✺✳✶❪ ❙❡❥❛ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡ X ✉♠❛ ❜❛s❡ ❞❡ V✳ ❊♥tã♦ T(V)∼=khXi ✭❝♦♠♦ á❧❣❡❜r❛s✮✳
❆ ➪❧❣❡❜r❛ ❙✐♠étr✐❝❛
❉❡✜♥✐çã♦ ✶✳✶✺ ✭➪❧❣❡❜r❛ ❙✐♠étr✐❝❛✮✳ ❙❡❥❛V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❉✐③❡♠♦s q✉❡(S(V), i)✱
♥❛ q✉❛❧ S(V) é ✉♠❛ á❧❣❡❜r❛ ❡ i: V −→T(V) ❧✐♥❡❛r✱ é ✉♠❛ á❧❣❡❜r❛ s✐♠étr✐❝❛ ❞❡ V s❡✱
♣❛r❛ t♦❞❛ á❧❣❡❜r❛ A ❡ t♦❞❛ f: V −→ A ❧✐♥❡❛r t❛❧ q✉❡ f(x)f(y) = f(y)f(x)✱ ♣❛r❛ t♦❞♦
✶✳ ❈♦♥❝❡✐t♦s Pr❡❧✐♠✐♥❛r❡s ✽
❞✐❛❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦✿
S(V)
V
A
¯
f i
f
❙❡♥❞♦ ❞❛❞❛ ♣♦r ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧✱ ✉♠❛ á❧❣❡❜r❛ s✐♠étr✐❝❛ ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧
V é ú♥✐❝❛ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✳ ❯♠ ♠♦❞❡❧♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✶✻✱ ❈❛♣ít✉❧♦ ■■❪
❝♦♠♦ ❛ s❡❣✉✐r✿
❙❡❥❛ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❉❡✜♥❛ S(V) = T(V)/I(V)✱ ❝♦♠ I(V) ♦ ✐❞❡❛❧ ❜✐❧❛t❡r❛❧
❣❡r❛❞♦ ♣♦r t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛x⊗y−y⊗x✱ ❝♦♠ x, y ∈V✳
✾
❈❛♣ít✉❧♦ ✷
❈♦á❧❣❡❜r❛s ❡ ❈♦♠ó❞✉❧♦s
◆❡st❡ ❝❛♣ít✉❧♦✱ ♥♦ss♦ ♦❜❥❡t✐✈♦ é ✐♥tr♦❞✉③✐r ❛s ❝♦á❧❣❡❜r❛s ❡ ♦s ❝♦♠ó❞✉❧♦s✿ ❉❡✜♥✐çõ❡s✱ ❡①❡♠♣❧♦s✱ ❤♦♠♦♠♦r✜s♠♦s ❡ ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s✳ ❊st❛s ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛s á❧❣❡❜r❛s ❡ ♦s ♠ó❞✉❧♦s sã♦ ♦ ❛❧✐❝❡r❝❡ ♣❛r❛ ♦ q✉❡ s❡rã♦ ❛s á❧❣❡❜r❛s ❞❡ ❍♦♣❢✳
❆s ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s ♣❛r❛ ❡st❡ ❝❛♣ít✉❧♦ sã♦ ♦s ❧✐✈r♦s ❬✶✵❪✱ ❬✶✻❪ ❡ ❬✷✵❪✳
✷✳✶ ➪❧❣❡❜r❛s
❆ ✜♠ ❞❡ ❢❛❝✐❧✐t❛r ❛ ✐♥tr♦❞✉çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦á❧❣❡❜r❛✱ ❝♦♥✈é♠ r❡✈✐s✐t❛r ❛ ❞❡✜♥✐çã♦ ❞❡ á❧❣❡❜r❛✳ P♦r ✐ss♦✱ tr❛③❡♠♦s ❛ s❡❣✉✐r✱ ✉♠❛ ❞❡✜♥✐çã♦ ❛❧t❡r♥❛t✐✈❛ ✉s❛♥❞♦ t❡♥s♦r❡s ❡ ❞✐❛❣r❛♠❛s✿
❉❡✜♥✐çã♦ ✷✳✶✳✶ ✭k✲á❧❣❡❜r❛✮✳ ❯♠❛ k✲á❧❣❡❜r❛ ✭❛ss♦❝✐❛t✐✈❛ ❝♦♠ ✉♥✐❞❛❞❡✮ é ✉♠❛ tr✐♣❧❛
(A, µ, η) ❡♠ q✉❡ A é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡ µ: A⊗A −→ A ❡ η: k −→ A sã♦ ❛♣❧✐❝❛çõ❡s
❧✐♥❡❛r❡s t❛✐s q✉❡ ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s sã♦ ❝♦♠✉t❛t✐✈♦s✿
A⊗A⊗A A⊗A
A⊗A A
id⊗µ
µ⊗id µ
µ
A⊗A
k⊗A A⊗k
A
µ η⊗id
∼
id⊗η
∼
❆ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛ s✐❣♥✐✜❝❛ q✉❡✱ ♣❛r❛ t♦❞♦ a, b, c ∈ A✱ (ab)c =
a(bc)✳ ❏á ♦ s❡❣✉♥❞♦ ❣❛r❛♥t❡ q✉❡✱ ♣❛r❛ t♦❞♦ a ∈ A ❡ ♣❛r❛ t♦❞♦ λ ∈ k✱ λa = (λ1A)a ❡
aλ=a(λ1A)✳
◆❛ ❞❡✜♥✐çã♦ ❛♥t❡r✐♦r✱ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛ é ❝❤❛♠❛❞❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡✳ ❆❧é♠ ❞✐ss♦✱ ♦ sí♠❜♦❧♦ ∼ ✐rá s❡♠♣r❡ r❡♣r❡s❡♥t❛r ✉♠ ❞♦s ✐s♦♠♦r✜s♠♦s
❝❛♥ô♥✐❝♦s ❞❛s Pr♦♣♦s✐çõ❡s ✶✳✼✱ ✶✳✽ ❡ ✶✳✾✳
❈♦♠♦ ❡♠ ❣❡r❛❧ ✉s❛r❡♠♦sk✲á❧❣❡❜r❛s ❛ ♠❡♥♦s q✉❡ s❡ ❞✐❣❛ ♦ ❝♦♥trár✐♦ ✐r❡♠♦s ❝❤❛♠á✲❧❛s
s✐♠♣❧❡s♠❡♥t❡ ❞❡ á❧❣❡❜r❛s✳
❊st❛ ♥♦✈❛ ✈❡rsã♦ ❞❛ ❞❡✜♥✐çã♦ ❞❡ á❧❣❡❜r❛ é ❡q✉✐✈❛❧❡♥t❡ à ❞❡✜♥✐çã♦ ✉s✉❛❧ ❞❡ á❧❣❡❜r❛ ✭❛ss♦❝✐❛t✐✈❛ ❝♦♠ ✉♥✐❞❛❞❡✮ ❝♦♠♦ ❡♠ ✶✳✶✵✳ ❉❡ ❢❛t♦✱ ❛❞♠✐t✐♥❞♦ ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ ❞❡✜♥❛ ♦ ♣r♦❞✉t♦a·b =µ(a⊗b)❡ 1A=η(1)✳ ❆ ❧✐♥❡❛r✐❞❛❞❡ ❞❡ µ❡ η ❡ ♦s ❞✐❛❣r❛♠❛s ✐rã♦ ❣❛r❛♥t✐r
♦ r❡s✉❧t❛❞♦✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❛❞♠✐t✐♥❞♦ ❛ ❞❡✜♥✐çã♦ ✉s✉❛❧✱ ❞❡✜♥❛µ(a⊗b) =a·b ❡η =φ✳
❆s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❞❡✜♥✐çã♦ ❝❧áss✐❝❛ ❞❡ á❧❣❡❜r❛ ❝♦♠ ♦ ❢❛t♦ ❞❡ φ s❡r ✉♠ ❤♦♠♦♠♦r✜s♠♦
✷✳ ❈♦á❧❣❡❜r❛s ❡ ❈♦♠ó❞✉❧♦s ✶✵
❈♦♠ ✐ss♦✱ ❞❛❞❛ ✉♠❛ á❧❣❡❜r❛ (A, µ, η)✱ ✐r❡♠♦s ❢r❡q✉❡♥t❡♠❡♥t❡ ❞❡♥♦t❛r µ(a⊗b) = ab
✭♥♦t❛çã♦ ❝❧áss✐❝❛ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✮ ❡η(1) = 1A ✭♥♦t❛çã♦ ❝❧áss✐❝❛ ❞❛ ✉♥✐❞❛❞❡✮✳
❖❜❡r✈❛♠♦s✱ ✜♥❛❧♠❡♥t❡✱ q✉❡ q✉❛♥❞♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ❡✈✐t❛r ❝♦♥❢✉sã♦✱ ♣♦❞❡♠♦s ✜①❛r ♦ sí♠❜♦❧♦ ❞❛ á❧❣❡❜r❛ ♥❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♦✉ ✉♥✐❞❛❞❡✿ µA, µB, ηA, ηB, . . .
❉❡✜♥✐çã♦ ✷✳✶✳✷ ✭➪❧❣❡❜r❛ ❈♦♠✉t❛t✐✈❛✮✳ ❯♠❛ á❧❣❡❜r❛ (A, µ, η) é ❞✐t❛ ❝♦♠✉t❛t✐✈❛ s❡ ♦
s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦✿
A⊗A
A⊗A
A
µ τ
µ
❡♠ q✉❡τ é ❛ ❛♣❧✐❝❛çã♦ ❞❡✜♥✐❞❛ ♥❛ ♣r♦♣♦s✐çã♦ ✶✳✻✳ ◆♦t❡ q✉❡ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❞✐❛❣r❛♠❛
❛❝✐♠❛ ❞✐③ q✉❡ab=ba♣❛r❛ t♦❞♦ a, b∈A✳
❉❡✜♥✐çã♦ ✷✳✶✳✸ ✭❍♦♠♦♠♦r✜s♠♦ ❞❡ ➪❧❣❡❜r❛s✮✳ ❙❡❥❛♠ (A, µA, ηA) ❡(B, µB, ηB)á❧❣❡❜r❛s
❡ f: A −→ B ❧✐♥❡❛r✳ ❉✐③❡♠♦s q✉❡ f é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s s❡ ♦s s❡❣✉✐♥t❡s
❞✐❛❣r❛♠❛s sã♦ ❝♦♠✉t❛t✐✈♦s✿
A⊗A B⊗B
A B
f⊗f
µA µB
f
A
k
B
f ηA
ηB
❖✉ s❡❥❛✱
f(a1a2) =f(a1)f(a2)
f(1A) = 1B
♣❛r❛ t♦❞♦sa1, a2 ∈A✳
❆❧é♠ ❞♦s ❡①❡♠♣❧♦s ❝♦♠✉♥s ❞❡ á❧❣❡❜r❛s ❝♦♠♦ ♦ ♣ró♣r✐♦ ❝♦r♣♦ k✱ á❧❣❡❜r❛s ❞❡ ♠❛tr✐③❡s✱
❞❡ ♣♦❧✐♥ô♠✐♦s✱ t❡♥s♦r✐❛❧✱ s✐♠étr✐❝❛✱ ❡t❝✳✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s ♦✉tr♦s ♣❛r❛ ✐❧✉str❛r ❛s ❞❡✜♥✐çõ❡s ❛❝✐♠❛✳
❊①❡♠♣❧♦ ✷✳✶✳✹✳ ❙❡❥❛(A, µ, η)✉♠❛ á❧❣❡❜r❛✳ ❉❡✜♥❛µop =µ◦τ✱ ♦✉ s❡❥❛✱µop(x⊗y) = yx✱
♣❛r❛ t♦❞♦x, y ∈A✳ ❊♥tã♦ (A, µop, η) é ✉♠❛ á❧❣❡❜r❛✱ ❝❤❛♠❛❞❛ á❧❣❡❜r❛ ♦♣♦st❛ ❞❡ A✳
❊①❡♠♣❧♦ ✷✳✶✳✺✳ ❙❡❥❛ M ✉♠ ♠♦♥♦✐❞❡ ❡ kM ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠ ❜❛s❡ {m | m ∈ M}
❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦ ❝♦♠❜✐♥❛çõ❡s ❧✐♥❡❛r❡s ✜♥✐t❛s ❞♦ t✐♣♦
X
m∈M
αmm
❡♠ q✉❡ (αm)m∈M é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❡❧❡♠❡♥t♦s ❡♠ k q✉❛s❡ t♦❞♦s ♥✉❧♦s✳ kM é ✉♠❛
á❧❣❡❜r❛ ❝♦♠ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛❞❛ ♣❡❧❛ ♦♣❡r❛çã♦ ❞♦ ♠♦♥♦✐❞❡ ❡ ❡st❡♥❞✐❞❛ ♣♦r ❧✐♥❡❛r✐❞❛❞❡✳ ❊①♣❧✐❝✐t❛♠❡♥t❡✱ s❡ x = Pm∈Mαmm ❡ y = Pn∈Mβnn ❡♥tã♦ xy = Pp∈Mγpp✱ ❡♠ q✉❡
γp =Pmn=pαmβn✳ ❆ ✉♥✐❞❛❞❡ ❞❡kM é ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞♦ ♠♦♥♦✐❞❡✳ ❆q✉✐ ✐♥t❡r❡ss❛✲♥♦s
✷✳ ❈♦á❧❣❡❜r❛s ❡ ❈♦♠ó❞✉❧♦s ✶✶
✷✳✷ ❈♦á❧❣❡❜r❛s
❆ ♥♦çã♦ ❞❡ ❝♦á❧❣❡❜r❛ é ♠♦t✐✈❛❞❛ ♣❡❧❛ ✐❞❡✐❛ ❞❛ ❞✉❛❧✐③❛çã♦ ❞❛s ❛♣❧✐❝❛çõ❡s ❞❡ ❞❡✜♥❡♠ ❛ ❡str✉t✉r❛ ❞❡ ✉♠❛ á❧❣❡❜r❛✱ r❡✈❡rt❡♥❞♦ ❛s ✢❡❝❤❛s✳ ❆ r❡❧❛çã♦ ❡♥tr❡ ❛s ❞✉❛s ❡str✉t✉r❛s ✜❝❛rá ♠❛✐s ❝❧❛r❛ ♥❛ ♣ró①✐♠❛ s❡çã♦✳
❉❡✜♥✐çã♦ ✷✳✷✳✶ ✭k✲❝♦á❧❣❡❜r❛✮✳ ❯♠❛ k✲❝♦á❧❣❡❜r❛ ✭❝♦❛ss♦❝✐❛t✐✈❛ ❝♦♠ ❝♦✉♥✐❞❛❞❡✮ é ✉♠❛
tr✐♣❧❛ (C,∆, ε) ❡♠ q✉❡ C é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ∆ : C −→ C ⊗ C ❡ ε: C −→ k sã♦
❛♣❧✐❝❛çõ❡s ❧✐♥❡❛r❡s t❛✐s q✉❡ ♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s sã♦ ❝♦♠✉t❛t✐✈♦s✿
C C⊗C
C⊗C C⊗C⊗C
∆
∆ id⊗∆
∆⊗id
C⊗C
k⊗C C⊗k
C
ε⊗id id⊗ε
∆
∼ ∼
❊♠ ❛♥❛❧♦❣✐❛ ❞✐r❡t❛ ❝♦♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ á❧❣❡❜r❛✱ ❝❤❛♠❛♠♦s∆❞❡ ❝♦♠✉❧t✐♣❧✐❝❛çã♦ ❡ε❞❡
❝♦✉♥✐❞❛❞❡ ❞❛k✲❝♦á❧❣❡❜r❛✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛ é ❝❤❛♠❛❞❛
❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡✳
❈♦♠♦ ❡♠ ❣❡r❛❧✱ ✉s❛r❡♠♦s k✲❝♦á❧❣❡❜r❛s ❛ ♠❡♥♦s q✉❡ s❡ ❞✐❣❛ ♦ ❝♦♥trár✐♦ ✐r❡♠♦s
❝❤❛♠á✲❧❛s s✐♠♣❧❡s♠❡♥t❡ ❞❡ ❝♦á❧❣❡❜r❛s✳
❉❡✜♥✐çã♦ ✷✳✷✳✷ ✭❈♦á❧❣❡❜r❛ ❈♦❝♦♠✉t❛t✐✈❛✮✳ ❯♠❛ ❝♦á❧❣❡❜r❛(C,∆, ε)é ❞✐t❛ ❝♦❝♦♠✉t❛t✐✈❛
s❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦✿
C⊗C
C
C⊗C
τ
∆
∆
❖ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦ ♠♦str❛ q✉❡ t♦❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♣♦ss✉✐ ❡str✉t✉r❛ ❞❡ ❝♦á❧❣❡❜r❛✳ ❊①❡♠♣❧♦ ✷✳✷✳✸✳ ❙❡❥❛S 6=∅ ✉♠ ❝♦♥❥✉♥t♦ ❡ kS ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠ ❜❛s❡ S✳ (kS,∆, ε)
é ✉♠❛ ❝♦á❧❣❡❜r❛ ❝♦♠ ❛ ❝♦♠✉❧t✐♣❧✐❝❛çã♦ ∆ : kS −→kS⊗kS ❞❛❞❛ ♣♦r ∆(s) =s⊗s ♣❛r❛
t♦❞♦s∈S ❡ ❝♦✉♥✐❞❛❞❡ε: kS−→k ❞❛❞❛ ♣♦rε(s) = 1 ♣❛r❛ t♦❞♦s∈S✳
❖s ♣ró①✐♠♦s ❞♦✐s ❡①❡♠♣❧♦s sã♦ ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s ❞♦ ❛♥t❡r✐♦r ❡ ♠❡r❡❝❡♠ s❡r ❛♣r❡s❡♥t❛❞♦s ♣♦rq✉❡ s✉r❣✐rã♦ ❞✐✈❡rs❛s ♦✉tr❛s ✈❡③❡s ♥♦ t❡①t♦✳
❊①❡♠♣❧♦ ✷✳✷✳✹✳ (k,∆, ε) é ✉♠❛ ❝♦á❧❣❡❜r❛ ❝♦♠ ❛ ❝♦♠✉❧t✐♣❧✐❝❛çã♦ ∆ : k −→k⊗k ❞❛❞❛
♣♦r∆(1) = 1⊗1❡ ❝♦✉♥✐❞❛❞❡ ε: k −→k ❞❛❞❛ ♣♦r ε(1) = 1✳
❊①❡♠♣❧♦ ✷✳✷✳✺✳ ❙❡❥❛M ✉♠ ♠♦♥♦✐❞❡ ❡kM ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠ ❜❛s❡ M✳ (kM,∆, ε)é
✉♠❛ ❝♦á❧❣❡❜r❛ ❝♦♠ ❛ ❝♦♠✉❧t✐♣❧✐❝❛çã♦ ∆ : kM −→ kM ⊗kM ❞❛❞❛ ♣♦r ∆(m) = m⊗m
♣❛r❛ t♦❞♦ m ∈M ❡ ❝♦✉♥✐❞❛❞❡ ε: kM −→ k ❞❛❞❛ ♣♦r ε(m) = 1 ♣❛r❛ t♦❞♦ m ∈ M✳ ❊♠
✷✳ ❈♦á❧❣❡❜r❛s ❡ ❈♦♠ó❞✉❧♦s ✶✷
❊①❡♠♣❧♦ ✷✳✷✳✻✳ ❙❡❥❛ (S,≤) ✉♠ ❝♦♥❥✉♥t♦ ♣❛r❝✐❛❧♠❡♥t❡ ♦r❞❡♥❛❞♦✳ ❯♠ ✐♥t❡r✈❛❧♦ ❡♠ S é
✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡S ♥ã♦ ✈❛③✐♦ ❞❛ ❢♦r♠❛
[x, y] :={z |x≤ z ≤y}.
❉✐③❡♠♦s q✉❡S é ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ s❡✱ t♦❞♦s s❡✉s ✐♥t❡r✈❛❧♦s sã♦ ✜♥✐t♦s✳ ❉❡♥♦t❡ ♦ ❝♦♥❥✉♥t♦
❞❡ ✐♥t❡r✈❛❧♦s ❞❡ S ♣♦r int(S)✳ ❙❡❥❛ V ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠ ❜❛s❡ int(S)✳ (V,∆, ε) é ✉♠❛
❝♦á❧❣❡❜r❛ ❝♦♠ ❛ ❝♦♠✉❧t✐♣❧✐❝❛çã♦ ∆ : V −→V ⊗V ❞❡✜♥✐❞❛ ❡♠ s✉❛ ❜❛s❡ int(S)♣♦r ∆([x, y]) = X
z∈[x,y]
[x, z]⊗[z, y]
❡ ❝♦✉♥✐❞❛❞❡ε: V −→k ❞❡✜♥✐❞❛ ♥❛ ❜❛s❡ ♣♦r
ε([x, y]) =δx,y.
❊①❡♠♣❧♦ ✷✳✷✳✼✳ ❙❡❥❛ n ≥ 1 ✉♠ ✐♥t❡✐r♦ ❡ Mc(n) ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦
n2✳ ❉❡♥♦t❡ ♣♦r (e
ij)1≤i,j≤n ✉♠❛ ❜❛s❡ ❞❡ M
c(n)✳ (Mc(n),∆, ε) é ✉♠❛ ❝♦á❧❣❡❜r❛ ❝♦♠ ❛
❝♦♠✉❧t✐♣❧✐❝❛çã♦∆ : Mc(n)−→Mc(n)⊗Mc(n) ❞❛❞❛ ♣♦r
∆(eij) = X
1≤p≤n
eip⊗epj
❡ ❝♦✉♥✐❞❛❞❡ ε: Mc(n) −→ k ❞❛❞❛ ♣♦r ε(e
ij) = δi,j. Mc(n) é ❝❤❛♠❛❞❛ ❞❡ ❝♦á❧❣❡❜r❛ ❞❡
♠❛tr✐③❡s✳
❊①❡♠♣❧♦ ✷✳✷✳✽✳ ❙❡❥❛(C,∆, ε)✉♠❛ ❝♦á❧❣❡❜r❛✳ (C,∆op, ε)é ✉♠❛ ❝♦á❧❣❡❜r❛✱ ❞✐t❛ ❝♦á❧❣❡❜r❛
♦♣♦st❛✱ ♥❛ q✉❛❧
∆op=τ ◦∆.
❆ ❝♦á❧❣❡❜r❛ ♦♣♦st❛ ❞❡ ✉♠❛ ❝♦á❧❣❡❜r❛C é ❞❡♥♦t❛❞❛ ♣♦rCcop✳
❆ss✐♠ ❝♦♠♦ é ❢❡✐t♦ ❝♦♠ á❧❣❡❜r❛s✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ ❡str✉t✉r❛ ❞❡ ❝♦á❧❣❡❜r❛ ♥♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❞❡ ❞✉❛s ❝♦á❧❣❡❜r❛s✿
❊①❡♠♣❧♦ ✷✳✷✳✾✳ ❙❡❥❛♠ (C,∆C, εC) ❡ (D,∆D, εD) ❞✉❛s ❝♦á❧❣❡❜r❛s✳ ❊♥tã♦ C⊗D t❡♠
❡str✉t✉r❛ ❞❡ ❝♦á❧❣❡❜r❛ ❞❛❞❛ ♣♦r
∆ = (id⊗τ⊗id)◦(∆C ⊗∆D)
❡
ε=ϕ◦εC ⊗εD,
❝♦♠ ϕ: k⊗k−→k ♦ ✐s♦♠♦r✜s♠♦ ❝❛♥ô♥✐❝♦✳
❙❡❥❛(C,∆, ε)✉♠❛ ❝♦á❧❣❡❜r❛✳ ■♥tr♦❞✉③✐♠♦s ❛ s❡❣✉✐♥t❡ ♥♦t❛çã♦✿ ∆1 = ∆,
∆2 = (∆⊗id)◦∆,
∆3 = (∆⊗id⊗id)◦∆2,
✳✳✳ ✳✳✳
∆n= (∆⊗idn−1)◦∆n−1.
✷✳ ❈♦á❧❣❡❜r❛s ❡ ❈♦♠ó❞✉❧♦s ✶✸
Pr♦♣♦s✐çã♦ ✷✳✷✳✶✵✳ ❙❡❥❛ (C,∆, ε) ✉♠❛ ❝♦á❧❣❡❜r❛✳ ❊♥tã♦ ♣❛r❛ t♦❞♦ n ≥ 2 ❡ t♦❞♦
p∈ {0, . . . , n−1}
∆n= idp⊗∆⊗idn−1−p◦∆n−1
❉❡♠♦♥str❛çã♦✳ ❋❛r❡♠♦s ♣♦r ✐♥❞✉çã♦ ❡♠ n✳ P❛r❛n = 2 t❡♠♦s ∆2 = (∆⊗id)◦∆ = (id⊗∆)◦∆
q✉❡ é s✐♠♣❧❡s♠❡♥t❡ ❛ ❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞❛ ❝♦♠✉❧t✐♣❧✐❝❛çã♦✳ ❆ss✉♠✐♥❞♦ q✉❡ ❛ ❛✜r♠❛çã♦ s❡❥❛ ✈á❧✐❞❛ ♣❛r❛n > 2✱ ❞❛❞♦ p∈ {1, . . . , n} t❡♠♦s
(idp⊗∆⊗idn−p)◦∆n = (idp⊗∆⊗idn−p)◦(idp−1⊗∆⊗idn−p)◦∆n−1
= (idp−1⊗((id⊗∆)◦∆)⊗idn−p)◦∆n−1
= (idp−1⊗((∆⊗id)◦∆)⊗idn−p)◦∆n−1
= (idp−1⊗∆⊗idn+1−p)◦(idp−1⊗∆⊗idn−p)◦∆n−1
= (idp−1⊗∆⊗idn−p+1)◦∆n.
◆♦t❡ q✉❡ ♣❛r❛ p = 0✱ t❡♠♦s ♣♦r ❞❡✜♥✐çã♦ q✉❡ ∆n+1 = (idp⊗∆⊗idn−p)◦∆n✱ ❧♦❣♦ ♣♦r
✐♥❞✉çã♦ ❡♠p✱ t❡♠♦s ♦ r❡s✉❧t❛❞♦ ♣❛r❛ t♦❞♦ p∈ {0, . . . , n}✳
❆♦ ❝♦♥trár✐♦ ❞♦ q✉❡ ♦❝♦rr❡ ♥❛ ♠✉❧t✐♣❧✐❝❛çã♦ q✉❡ ❧❡✈❛ ❞♦✐s ❡❧❡♠❡♥t♦s ❡♠ ✉♠ só✱ ❛ ❝♦♠✉❧t✐♣❧✐❝❛çã♦ ❧❡✈❛ ✉♠ ❡❧❡♠❡♥t♦ ❡♠ ✉♠❛ s♦♠❛ ✜♥✐t❛ ❞❡ t❡♥s♦r❡s ❞❡ ❡❧❡♠❡♥t♦s✱ ♦ q✉❡ ♣♦❞❡ t♦r♥❛r ♦s ❝á❧❝✉❧♦s ✉♠ t❛♥t♦ ❝♦♠♣❧✐❝❛❞♦s✳ P❛r❛ ❢❛❝✐❧✐t❛r✱ ✉s❛r❡♠♦s ✉♠❛ ♥♦t❛çã♦ ♣❛r❛ ❛ ❝♦♠✉❧t✐♣❧✐❝❛çã♦ ❝❤❛♠❛❞❛ ❞❡ ◆♦t❛çã♦ ❞❡ ❙✇❡❡❞❧❡r ✭♦✉ ◆♦t❛çã♦ ❙✐❣♠❛✮✿ ❉❛❞❛ ✉♠❛ ❝♦á❧❣❡❜r❛ (C,∆, ε) ❞❡♥♦t❛r❡♠♦s
∆(c) =
n X
i=1
ci1⊗ci2
♣♦r
∆(c) =Xc1⊗c2.
❈♦♠ ✐ss♦✱ ♣❡❧❛ ❝♦❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❣❡♥❡r❛❧✐③❛❞❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✱ ♣❛r❛n≥1✱ ∆n(c) =
X
c1⊗. . .⊗cn+1.
❯s❛♥❞♦ ❡ss❛ ♥♦t❛çã♦ ♣♦❞❡♠♦s t❛♠❜é♠ r❡❡s❝r❡✈❡r ♦ s❡❣✉♥❞♦ ❞✐❛❣r❛♠❛ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦á❧❣❡❜r❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
X
ε(c1)c2 =
X
c1ε(c2) =c.
❉❡✜♥✐çã♦ ✷✳✷✳✶✶ ✭❍♦♠♦♠♦r✜s♠♦ ❞❡ ❈♦á❧❣❡❜r❛s✮✳ ❙❡❥❛♠ (C,∆C, εC) ❡ (D,∆D, εD)
❝♦á❧❣❡❜r❛s ❡ g: C −→ D ❧✐♥❡❛r✳ ❉✐③❡♠♦s q✉❡ g é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❝♦á❧❣❡❜r❛s s❡
♦s s❡❣✉✐♥t❡s ❞✐❛❣r❛♠❛s sã♦ ❝♦♠✉t❛t✐✈♦s✿
C D
C⊗C D⊗D
g
∆C ∆D
g⊗g
D
C
k
εD
g
εC
P❡❧❛ ♥♦t❛çã♦ ❞❡ ❙✇❡❡❞❧❡r✱ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ♣r✐♠❡✐r♦ ❞✐❛❣r❛♠❛ ❛❝✐♠❛ ♣♦❞❡ s❡r ❡s❝r✐t❛✱ ♣❛r❛ t♦❞♦ c∈C✱ ❝♦♠♦✿
X
g(c)1⊗g(c)2 =
X
✷✳ ❈♦á❧❣❡❜r❛s ❡ ❈♦♠ó❞✉❧♦s ✶✹
✷✳✸ ❉✉❛❧✐❞❛❞❡ ❡♥tr❡ ➪❧❣❡❜r❛s ❡ ❈♦á❧❣❡❜r❛s
❉❛❞♦ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ❝♦♥s✐❞❡r❡ V∗ = Hom(V, k) ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞✉❛❧ ❛ V✳ ❙❡
ϕ: V −→ W é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✱ ❞❡✜♥✐♠♦s ϕ∗: W∗ −→ V∗ ♣♦r
ϕ∗(f)(v) = f(ϕ(v))♣❛r❛ t♦❞♦f ∈W∗ ❡v ∈V✳ ❊ss❛s ✐♥❢♦r♠❛çõ❡s s❡rã♦ ♥❡❝❡ssár✐❛s ♥❡ss❛
s❡çã♦ ♣❛r❛ ♠♦str❛r q✉❡✱ ❞❛❞❛ ✉♠❛ ❝♦á❧❣❡❜r❛ C✱ ❡♥tã♦ C∗ ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ♥❛t✉r❛❧
❞❡ á❧❣❡❜r❛✱ ♣♦r ❞✉❛❧✐❞❛❞❡✳ ❈♦♠❡❝❡♠♦s ❝♦♠ ❛❧❣✉♥s ❧❡♠❛s✿
▲❡♠❛ ✷✳✸✳✶✳ ❙❡❥❛♠ V ❡ W ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❡ t ∈ V ⊗W✱ ❝♦♠ t 6= 0✳ ❊♥tã♦ ❡①✐st❡
n >0 t❛❧ q✉❡
t =
n X
i=1
vi⊗wi,
❝♦♠ vi ❡ wi ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ♣❛r❛ t♦❞♦ i= 1, ...n✳
❉❡♠♦♥str❛çã♦✳ ➱ ❝❧❛r♦ q✉❡ s❡♠♣r❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡rt ❝♦♠ t♦❞♦s ❝♦♠♣♦♥❡♥t❡s vi ♦✉wi
❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ♣❛r❛ ❛❧❣✉♠n✳ ❋✐①❡ n ❝♦♠♦ ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ t❛❧ q✉❡ t
s❡ ❡s❝r❡✈❛ ❞❛ ❢♦r♠❛
t =
n X
i=1
vi⊗wi,
❝♦♠ ♦s vi ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ▼♦str❡♠♦s q✉❡ ♦s wi t❛♠❜é♠ sã♦ ❧✐♥❡❛r♠❡♥t❡
✐♥❞❡♣❡♥❞❡♥t❡s✳ ❉❡ ❢❛t♦✱ s❡ ♥ã♦ ♦ ❢♦ss❡✱ t❡rí❛♠♦s✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ q✉❡wn s❡r✐❛
❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ❞❡♠❛✐s✿
wn = n−1
X
i=1
αiwi.
❈♦♠ ✐ss♦✱
t=
n−1
X
i=1
vi⊗wi+vn⊗wn
=
n−1
X
i=1
vi⊗wi+vn⊗ n−1
X
i=1
αiwi !
=
n−1
X
i=1
(vi+αivn)⊗wi
▼❛s é ❝❧❛r♦ q✉❡ {v1 +α1vn, . . . , vn−1 +αn−1vn} é ❧✐♥❡r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡✱ ♣♦ré♠ ❝♦♠
✐ss♦ t❡♠♦s ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞❡t ❝♦♠ n−1 ❝♦♠♣♦♥❡♥t❡svi ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱
♦ q✉❡ é ✉♠ ❛❜s✉r❞♦ ✭♣❡❧❛ ♠✐♥✐♠❛❧✐❞❛❞❡ ❞❡n✮✳
▲❡♠❛ ✷✳✸✳✷✳ ❙❡❥❛♠ M ❡ V ❡s♣❛ç♦s ✈❡t♦r✐❛s ❡ ϕ: M∗⊗V −→Hom(M, V) ❧✐♥❡❛r ❞❛❞❛
♣♦r
ϕ(f⊗v)(m) = f(m)v,
♣❛r❛ t♦❞♦sf ∈M∗, v ∈V ❡ m∈M✳ ❊♥tã♦ϕ é ✐♥❥❡t♦r❛✳ ❆❧é♠ ❞✐ss♦✱ s❡M t✐✈❡r ❞✐♠❡♥sã♦
✜♥✐t❛ ❡♥tã♦ ϕ é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛x∈M∗⊗V t❛❧ q✉❡ φ(x) = 0✳ P❡❧♦ ▲❡♠❛ ✷✳✸✳✶✱ ❡①✐st❡ p >0✱t❛❧ q✉❡
x=
p X
i=1
✷✳ ❈♦á❧❣❡❜r❛s ❡ ❈♦♠ó❞✉❧♦s ✶✺
❝♦♠ fi ∈M∗ ❡ vi ∈V ❧✐♥❡r❛♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❈♦♠ ✐ss♦✱ ♣❛r❛ t♦❞♦ m∈M✱
0 =φ(x)(m) =φ
p X
i=1
fi⊗vi !
(m) =
p X
i=1
φ(fi⊗vi)(m) = p X
i=1
fi(m)vi.
❈♦♠♦ ♦s vi sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ t❡♠♦s q✉❡ fi(m) = 0 ♣❛r❛ t♦❞♦ i= 1, . . . , p ❡
♣❛r❛ t♦❞♦m∈M✳ P♦rt❛♥t♦ x= 0✳
❆❣♦r❛✱ ❛❞♠✐t✐♥❞♦ q✉❡ M t❡♥❤❛ ❞✐♠❡♥sã♦ ✜♥✐t❛ n✱ t♦♠❡{m1, . . . , mn}✉♠❛ ❜❛s❡ ♣❛r❛
M ❡ s❡❥❛ {m∗
1, . . . , m∗n} ✉♠❛ ❜❛s❡ ❞✉❛❧ ♣❛r❛ M∗✱ ♦✉ s❡❥❛✱ m∗i(mj) = δij✱ ♣❛r❛ t♦❞♦
1≤i, j ≤n✳ ➱ ❝❧❛r♦ q✉❡✱ ♣❛r❛ t♦❞♦m ∈M✱
m =
n X
i=1
m∗i(m)mi.
❆ss✐♠✱ ❞❛❞♦ f ∈Hom(M, V)✱
f(m) =f
n X
i=1
m∗
i(m)mi ! = n X i=1 m∗
i(m)f(mi).
▲♦❣♦✱ t♦♠❛♥❞♦
x=
n X
i=1
m∗if(mi)
t❡r❡♠♦s✱ ♣❛r❛ t♦❞♦m ∈M✱
φ(x)(m) = φ
n X
i=1
m∗
if(mi) !
(m) =
n X
i=1
m∗
i(m)f(mi) = f(m).
P♦rt❛♥t♦φ é s♦❜r❡❥❡t♦r❛✱ ❡ ❛ss✐♠ ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳
▲❡♠❛ ✷✳✸✳✸✳ ❙❡❥❛♠ M ❡ N ❡s♣❛ç♦s ✈❡t♦r✐❛s ❡ ψ: Hom(M, N∗)−→(M⊗N)∗ ❧✐♥❡❛r
❞❛❞❛ ♣♦r
ψ(g)(m⊗n) =g(m)(n),
♣❛r❛ t♦❞♦s g ∈ Hom(M, N∗), m ∈ M ❡ n ∈ N✳ ❊♥tã♦ ψ é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ ❡s♣❛ç♦s
✈❡t♦r✐❛✐s✳
❉❡♠♦♥str❛çã♦✳ P❛r❛ ♠♦str❛r ❡ss❡ r❡s✉❧t❛❞♦ r❡❝♦r❞❡♠♦s ♦ s❡❣✉✐♥t❡ ✐s♦♠♦r✜s♠♦ ❝❛♥ô♥✐❝♦ ❡♥✈♦❧✈❡♥❞♦ ♦ ♦♣❡r❛❞♦rHom❡ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧✿
❙❡❥❛♠ A, B ❡ C ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❡♥tã♦
Hom(A⊗B, C)∼= Hom(A,Hom(B, C)).
❯s❛♥❞♦ ❡ss❡ ❢❛t♦ ♦ ✐s♦♠♦r✜s♠♦ t❡♠♦s q✉❡Hom(M, N∗)❡(M ⊗N)∗ sã♦ ✐s♦♠♦r❢♦s ♣♦✐s
Hom(M, N∗) = Hom (M,Hom(N, k))
∼
= Hom (M ⊗N, k) = (M ⊗N)∗.
➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ ❡st❡ ✐s♦♠♦r✜s♠♦ ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ❛♣❧✐❝❛çã♦ψ ❞♦ ❡♥✉♥❝✐❛❞♦ ✭♦ q✉❡
✷✳ ❈♦á❧❣❡❜r❛s ❡ ❈♦♠ó❞✉❧♦s ✶✻
Pr♦♣♦s✐çã♦ ✷✳✸✳✹✳ ❙❡❥❛♠ M ❡ N ❡s♣❛ç♦s ✈❡t♦r✐❛s ❡ ρ:M∗⊗N∗ −→(M ⊗N)∗ ❧✐♥❡❛r
❞❛❞❛ ♣♦r
ρ(f ⊗g)(m⊗n) = f(m)g(n),
♣❛r❛ t♦❞♦sf ∈M∗, g ∈N∗ ❡n∈N✳ ❊♥tã♦ρé ✐♥❥❡t♦r❛✳ ❆❧é♠ ❞✐ss♦✱ s❡M t✐✈❡r ❞✐♠❡♥sã♦
✜♥✐t❛ ❡♥tã♦ ρ é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳
❉❡♠♦♥str❛çã♦✳ ❇❛st❛ ♦❜s❡r✈❛r q✉❡ρ=ψ◦ϕ0✱ ❡♠ q✉❡φ0 é ❛ ❛♣❧✐❝❛çã♦ φ ❞♦ ▲❡♠❛ ✷✳✸✳✷
❝♦♠ V =N∗ ❡ ψ é ❛ ❛♣❧✐❝❛çã♦ ❞♦ ▲❡♠❛ ✷✳✸✳✸✳
❈♦r♦❧ár✐♦ ✷✳✸✳✺✳ ❙❡❥❛♠ V1, . . . , Vn ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ ❊♥tã♦ ❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r
θ: V∗
1 ⊗. . .⊗Vn∗ −→(V1⊗. . .⊗Vn)∗ ❞❛❞❛ ♣♦r
θ(f1⊗. . .⊗fn)(v1⊗. . .⊗vn) =f1(v1). . . fn(vn)
é ✐♥❥❡t♦r❛✳ ▼❛✐s ❛✐♥❞❛✱ s❡ Vi é ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ♣❛r❛ t♦❞♦ i = 1, . . . n ❡♥tã♦ θ é ✉♠
✐s♦♠♦r✜s♠♦✳
❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ❝♦r♦❧ár✐♦ é ✐♠❡❞✐❛t❛ ❛ ♣❛rt✐r ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✳ ❚❡♦r❡♠❛ ✷✳✸✳✻✳ ❙❡ (C,∆, ε) é ✉♠❛ ❝♦á❧❣❡❜r❛ ❡♥tã♦ C∗ ♣♦ss✉✐ ❡str✉t✉r❛ ❞❡ á❧❣❡❜r❛✳
❉❡♠♦♥str❛çã♦✳ ❚♦♠❡ρ♥❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ❝♦♠M =N =C✱ ♦✉ s❡❥❛✱ ♦ ❤♦♠♦♠♦r✜s♠♦
✐♥❥❡t♦r ρ: C∗⊗C∗ −→(C⊗C)∗✳ ❈♦♥s✐❞❡r❡ t❛♠❜é♠ ♦ ✐s♦♠♦r✜s♠♦ ϕ: k −→k∗✱ t❛❧ q✉❡
ϕ(1) = id✳ ❉❡✜♥❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ❛ ✉♥✐❞❛❞❡ ❡♠C∗ ♣♦r
µ= ∆∗◦ρ ❡ η =ε∗◦ϕ
◆♦t❡ q✉❡✱ ❝♦♠♦∆∗(h) =h◦∆♣❛r❛ t♦❞♦h ∈(C⊗C)∗✱ ❡♥tã♦
µ(f⊗g)(c) = ∆∗◦ρ(f⊗g)(c) =ρ(f⊗f)◦∆(c) =ρ(f⊗g)Xc1⊗c2
=Xf(c1)g(c2)
♣❛r❛ t♦❞♦f, g∈C∗ ❡c∈C✳ ❊ss❛ ♠✉❧t✐♣❧✐❝❛çã♦ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ♣r♦❞✉t♦ ❞❡ ❝♦♥✈♦❧✉çã♦
❡ ❞❡♥♦t❛✲s❡ µ(f ⊗g) ♣♦r f ∗g✳ ✭▼❛✐s ❛❞✐❛♥t❡✱ ♥❛ s❡çã♦ ✸✳✷ ✈♦❧t❛r❡♠♦s ❛ ❢❛❧❛r s♦❜r❡ ♦
♣r♦❞✉t♦ ❞❡ ❝♦♥✈♦❧✉çã♦ ❞❡ ✉♠❛ ❢♦r♠❛ ♠❛✐s ❛♠♣❧❛✮✳
◆♦t❡ t❛♠❜é♠ q✉❡✱ ❝♦♠♦ ε∗(h) = h◦ε✱ ♣❛r❛ t♦❞♦ h∈k∗✱ ❡♥tã♦
η(1)(c) =ε∗(φ(1))(c) = (id◦ε)(c) =ε(c)
♣❛r❛ t♦❞♦c∈C✳ ▲♦❣♦ η(α)(c) =αη(1)(c)♣❛r❛ t♦❞♦ α∈k ❡c∈C✳
▼♦str❡♠♦s ❛❣♦r❛ q✉❡ (C∗, µ, η) é✱ ❞❡ ❢❛t♦✱ ✉♠❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❡ ✉♥✐tár✐❛✿
• µ é ❛ss♦❝✐❛t✐✈❛ ♣♦✐s✱ ♣❛r❛ t♦❞♦c∈C ❡f, g ∈C∗✱
((f∗g)∗h)(c) = X(f ∗g)(c1)h(c2)
=Xf(c1)g(c2)h(c3)
=Xf(c1)(g∗h)(c2)
= (f ∗(g∗h))(c),
