Fórmula de caractere para álgebras de Lie Semissimples de dimensão finita

(1)

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❋ór♠✉❧❛ ❞❡ ❈❛r❛❝t❡r❡ ♣❛r❛

➪❧❣❡❜r❛s ❞❡ ▲✐❡ ❙❡♠✐ss✐♠♣❧❡s ❞❡

❉✐♠❡♥sã♦ ❋✐♥✐t❛

●✉st❛✈♦ P❡r❡✐r❛ ●♦♠❡s

(2)

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❋ór♠✉❧❛ ❞❡ ❈❛r❛❝t❡r❡ ♣❛r❛

➪❧❣❡❜r❛s ❞❡ ▲✐❡ ❙❡♠✐ss✐♠♣❧❡s ❞❡

❉✐♠❡♥sã♦ ❋✐♥✐t❛

❉✐s❝❡♥t❡✿ ●✉st❛✈♦ P❡r❡✐r❛ ●♦♠❡s ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❆♥❞ré ●✐♠❡♥❡③ ❇✉❡♥♦

❉✐ss❡rt❛çã♦ ♦r✐❡♥t❛❞❛ ♣❡❧♦ Pr♦❢✳ ❉r✳ ❆♥❞ré ●✐♠❡♥❡③ ❇✉❡♥♦ ❡ ❛♣r❡✲ s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞♦ ♠❡str❛❞♦ ❡♠ ▼❛t❡♠á✲ t✐❝❛✳

(3)

✐✐✐

❆❣r❛❞❡❝✐♠❡♥t♦s

❆ ❉❡✉s ♣❡❧❛s ♦♣♦rt✉♥✐❞❛❞❡s✳

❆♦s ♠❡✉s ♣❛✐s ❈❧❡❛♠ár❝✐❛ ❡ ❋❛❜✐❛♥♦✱ ♠❡✉s ✐r♠ã♦s ❈❛♠✐❧❛ ❡ ❱✐❝t♦r ❡ ❢❛♠✐❧✐✲ ❛r❡s ♣❡❧♦ ❝❛r✐♥❤♦ ❡ ❛♣♦✐♦✳

❆ ♠✐♥❤❛ ♥❛♠♦r❛❞❛ ●❛❜r✐❡❧❧❡ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ❛♠♦r✳ ❆♦s ❝♦❧❡❣❛s✱ ❛♠✐❣♦s ❡ ♣r♦❢❡ss♦r❡s ❞❛ ❯❋▼●✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢❡ss♦r ❆♥❞ré✱ q✉❡ ✜③❡r❛♠ ♣❛rt❡ ❞❡st❛ ❝❛♠✐♥❤❛❞❛✳

❆♦s ♠❡✉s ❛♥t✐❣♦s ♣r♦❢❡ss♦r❡s✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ♣r♦❢❡ss♦r❡s ❘♦s✐✈❛❧❞♦ ❡ ❙❡❜❛s✲ t✐ã♦✱ q✉❡ ❛❝r❡❞✐t❛r❛♠ ♥♦ ♠❡✉ ♣♦t❡♥❝✐❛❧✳

(4)

✐✈

❘❡s✉♠♦

❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ tr❛❜❛❧❤♦ é ❞❡s❝r❡✈❡r ❛s r❡♣r❡s❡♥t❛çõ❡s ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡ s❡♠✐ss✐♠♣❧❡s g s♦❜r❡ _C ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛✱ ♦♥❞❡ g t❛♠❜é♠

t❡♠ ❞✐♠❡♥sã♦ ✜♥✐t❛✳ ■♥✐❝✐❛❧♠❡♥t❡✱ é ♥❡❝❡ssár✐♦ ♦ ❡st✉❞♦ ❞❛s s✉❜á❧❣❡❜r❛s ❞❡ ❈❛rt❛♥✱ ❥✉♥t❛♠❡♥t❡ ❞❛ t❡♦r✐❛ ❞❡ r❛í③❡s✱ q✉❡ ♥♦s ❧❡✈❛ à s❡❣✉✐♥t❡ ❞❡❝♦♠♣♦s✐çã♦✿

g=h⊕M

α∈R gα,

♦♥❞❡ h ⊂ g é ✉♠❛ s✉❜á❧❣❡❜r❛ ❞❡ ❈❛rt❛♥✳ ❆♦ ♠❡s♠♦ t❡♠♣♦✱ ❛s ❢ór♠✉✲

❧❛s ❞❡ ❋r❡✉❞❡♥t❤❛❧ ❡ ❲❡②❧ ♥♦s ♠♦str❛ ❛s ❞✐♠❡♥sõ❡s ❞❡st❛s r❡♣r❡s❡♥t❛çõ❡s ❝✐t❛❞❛s ❛❝✐♠❛✳ ❆❧é♠ ❞✐ss♦✱ ❛♣r❡s❡♥t❛♠♦s ❛ t❡♦r✐❛ ❞❡ ❝❛r❛❝t❡r❡ ❡ ❛♥❡❧ ❞❡ r❡✲ ♣r❡s❡♥t❛çõ❡s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ♦❜t❡r ❢❡rr❛♠❡♥t❛s q✉❡ ❛✉①✐❧✐❛♠ ♥♦ ❡st✉❞♦ ❞❛s r❡♣r❡s❡♥t❛çõ❡s ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡✳

(5)

✈

❆❜str❛❝t

❚❤❡ ❣♦❛❧ ♦❢ t❤✐s ✇♦r❦ ✐s ❞❡s❝r✐❜❡ t❤❡ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ s❡♠✐s✐♠♣❧❡ ▲✐❡ ❛❧❣❡❜r❛sg ♦✈❡rC✱ ✇❤❡r❡g ❛❧s♦ ❤❛s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥✳ ■♥✐t✐✲

❛❧❧②✱ ✐s ♥❡❝❡ss❛r② t❤❡ st✉❞② ♦❢ ❈❛rt❛♥ s✉❜❛❧❣❡❜r❛s✱ t♦❣❡t❤❡r ♦❢ r♦♦t s②st❡♠s✱ ✇❤✐❝❤ ❧❡❛❞s t♦ ❞❡❝♦♠♣♦s✐t✐♦♥✿

g=h⊕M

α∈R gα,

✇❤❡r❡ h ⊂ g ✐s ❛ ❈❛rt❛♥ s✉❜❛❧❣❡❜r❛✳ ❆t t❤❡ s❛♠❡ t✐♠❡✱ ❋r❡✉❞❡♥t❤❛❧✬s ❛♥❞

❲❡②❧✬s ❢♦r♠✉❧❛s ❣✐✈❡ ✉s t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡s❡ r❡♣r❡s❡♥t❛t✐♦♥s ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✳ ▼♦r❡♦✈❡r✱ ✇❡ ♣r❡s❡♥t t❤❡ t❤❡♦r② ♦❢ ❝❤❛r❛❝t❡rs ❛♥❞ r❡♣r❡s❡♥t❛t✐♦♥ r✐♥❣ ❛s t♦♦❧s t❤❛t ❤❡❧♣ ✉s ✉♥❞❡rst❛♥❞ ♦❢ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ▲✐❡ ❛❧❣❡❜r❛s✳

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✈✐

■♥tr♦❞✉çã♦

❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r ❛s á❧❣❡❜r❛s ❞❡ ▲✐❡ s❡♠✐ss✐♠♣❧❡s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ s♦❜r❡ C✱ ❜❡♠ ❝♦♠♦ s✉❛s r❡♣r❡s❡♥t❛çõ❡s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛✳ ❉❡st❡ ♠♦❞♦✱ ♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ♠❡♥❝✐♦♥❛❞♦ ♥❡st❛ ❞✐ss❡rt❛çã♦ é ❛ ♦❜t❡♥çã♦ ❞❛ ❋ór♠✉❧❛ ❞❡ ❲❡②❧ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ❞✐♠❡♥sã♦ ❞❡st❛s r❡♣r❡s❡♥t❛çõ❡s ❝✐t❛❞❛s ❛❝✐♠❛✳

❖ ú♥✐❝♦ ♣ré✲r❡q✉✐s✐t♦ ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❛ ➪❧❣❡❜r❛ ▲✐♥❡❛r ❡ ▼✉❧t✐❧✐♥❡❛r✳ P♦r ❡①❡♠♣❧♦✱ ✉s❛♠♦s ❝♦♥st❛♥t❡♠❡♥t❡ ♦s ❝♦♥❝❡✐t♦s ❞❡ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧✱ ❛✉t♦✈❛❧♦r✱ ❛✉t♦✈❡t♦s ❡ ❛✉t♦❡s♣❛ç♦s✳ ❚♦❞❛✈✐❛✱ ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ✉s❛❞♦s ❞❛ ➪❧❣❡❜r❛ ▲✐♥❡❛r ❡ ▼✉❧t✐❧✐♥❡❛r s❡rã♦ ❝✐t❛❞♦s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✳

❊st❡ tr❛❜❛❧❤♦ ❢♦✐ ❞✐✈✐❞✐❞♦ ❡♠ ♥♦✈❡ s❡çõ❡s✳ ❖s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❛s á❧❣❡❜r❛s ❞❡ ▲✐❡ ❡stã♦ ♣r❡s❡♥t❡s ♥❛ ❙❡çã♦ ✶✱ ❜❡♠ ❝♦♠♦ ♦s ❚❡♦r❡♠❛s ❞❡ ❊♥❣❡❧ ❡ ▲✐❡ s♦❜r❡ á❧❣❡❜r❛s ❞❡ ▲✐❡ ♥✐❧♣♦t❡♥t❡s ❡ s♦❧ú✈❡✐s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ q✉❡ sã♦ t❡♦r❡♠❛s ✐♥❞✐s♣❡♥sá✈❡✐s ♥❛ t❡♦r✐❛ ❞❛s á❧❣❡❜r❛s ❞❡ ▲✐❡✳ ❱❛❧❡ r❡ss❛❧t❛r q✉❡ ❡st❛ s❡çã♦ ❢♦✐ ❜❛s❡❛❞❛ ♥♦ ❧✐✈r♦ ■♥tr♦❞✉❝t✐♦♥ t♦ ▲✐❡ ❛❧❣❡❜r❛s ❛♥❞ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ❞♦ ❛✉t♦r ❏❛♠❡s ❊✳ ❍✉♠♣❤r❡②s ✭r❡❢❡rê♥❝✐❛ ❬✷❪✮✳

❏á ♥❛ ❙❡çã♦ ✷✱ ♠❡♥❝✐♦♥❛♠♦s ✉♠ ❝r✐tér✐♦ ❞❡ s❡♠✐ss✐♠♣❧✐❝✐❞❛❞❡ ♣❛r❛ á❧❣❡❜r❛s ❞❡ ▲✐❡ ❡ ❛♦ ♠❡s♠♦ t❡♠♣♦✱ ❛❜♦r❞❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❲❡②❧✱ ❢❛③❡♠♦s ✉♠❛ ❜r❡✈❡ ❞✐s❝✉ssã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❆❞♦ ❡ ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡st❡ t❡♦r❡♠❛ q✉❛♥❞♦ ❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ é s❡♠✐ss✐♠♣❧❡s✳

❈♦♠❡ç❛♠♦s ❛ ❙❡çã♦ ✸ ❞❡✜♥✐♥❞♦ ❛s s✉❜á❧❣❡❜r❛s ❞❡ ❈❛rt❛♥ ❡ ❞❡♠♦♥s✲ tr❛♠♦s q✉❡ t♦❞❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ s♦❜r❡ C♣♦ss✉✐ s✉❜á❧❣❡❜r❛ ❞❡ ❈❛rt❛♥ ✉s❛♥❞♦ ❛ ❚♦♣♦❧♦❣✐❛ ❞❡ ❩❛r✐s❦✐ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ❞❡♠♦♥str❛t✐✈❛✳ ❚❛♠❜é♠✱ ❛❜♦r❞❛♠♦s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ❛ r❡s♣❡✐t♦ ❞❛s s✉❜á❧❣❡❜r❛s ❞❡ ❈❛rt❛♥ q✉❛♥❞♦ ❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ é s❡♠✐ss✐♠♣❧❡s✳

❖ ♦❜❥❡t✐✈♦ ❞❛ ❙❡çã♦ ✹ é ❝❛r❛❝t❡r✐③❛r ❛s r❡♣r❡s❡♥t❛çõ❡s ✭♠ó❞✉❧♦s✮ ❞❡

sl2(C) ❡ s✐st❡♠❛s ❞❡ r❛í③❡s✱ ♠❛tr✐③❡s ❞❡ ❈❛rt❛♥✱ ❞✐❛❣r❛♠❛s ❞❡ ❉②♥❦✐♥ ❡stã♦

♣r❡s❡♥t❡s ♥❛ ❙❡çã♦ ✺✳

◆❛ ❙❡çã♦ ✻✱ ✜③❡♠♦s ✉♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ ❛s á❧❣❡❜r❛s ❞❡ ▲✐❡ s❡♠✐ss✐♠✲ ♣❧❡s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ s♦❜r❡C❡ ✉♠ ❞♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ♣r❡s❡♥t❡s ♥❡st❛ s❡çã♦ é q✉❡ t♦❞❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ♥❡st❛s ❝♦♥❞✐çõ❡s ♣♦ss✉✐ ❛ ❞❡❝♦♠♣♦s✐çã♦

g=h⊕M

α∈R gα

❡♠ ❛✉t♦❡s♣❛ç♦s gα ={X ∈g | ❛❞(H)X = [H, X] =α(H)X, ∀H ∈h}✱ ♦♥❞❡

(7)

✈✐✐

❆s r❡♣r❡s❡♥t❛çõ❡s ❞❛s á❧❣❡❜r❛s ❞❡ ▲✐❡ s❡♠✐ss✐♠♣❧❡s ❡ ❛ t❡♦r✐❛ ❞❡ ♣❡s♦s sã♦ ❞❡s❝r✐t❛s ♥❛ ❙❡çã♦ ✼✳ ❉❡st❛❝❛♠♦s q✉❡ ❛s ❙❡çõ❡s ✷ ❛té ✼ ❢♦r❛♠ ❜❛s❡❛❞♦s ♥♦ ❧✐✈r♦ ❈♦♠♣❧❡① ❙❡♠✐s✐♠♣❧❡ ▲✐❡ ❆❧❣❡❜r❛s ❞♦ ❛✉t♦r ❏✳P✳ ❙❡rr❡ ✭r❡❢❡rê♥❝✐❛ ❬✹❪✮✳ ❆ ❙❡çã♦ ✽ ❝♦♥té♠ ♦s r❡s✉❧t❛❞♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❡st❛ ❞✐ss❡rt❛çã♦✳ P♦r ❡①❡♠♣❧♦✱ ♥❡❧❛ ❡♥❝♦♥tr❛♠♦s ❛s ❝â♠❛r❛s ❞❡ ❲❡②❧✱ ♦ ❡st✉❞♦ ❞♦s ❝❛r❛❝t❡r❡s ❞❛s r❡♣r❡❡♥t❛çõ❡s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ❡ ♦ ❛♥❡❧ ❞❡ r❡♣r❡s❡♥t❛çõ❡s✳ ❱❛❧❡ ❞❡st❛❝❛r q✉❡ ♦ ❛♥❡❧ ❞❡ r❡♣r❡s❡♥t❛çõ❡s é ✉♠ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ♦♥❞❡ ❛s ✈❛r✐á✈❡✐s sã♦ ❛s ❝❧❛ss❡s ❞❛s r❡♣r❡s❡♥t❛çõ❡s ✐rr❡❞✉tí✈❡✐s ❞❡ ♣❡s♦ ♠❛✐s ❛❧t♦ω1, . . . , ωn✱ ❝❤❛♠❛❞♦s

♣❡s♦s ❢✉♥❞❛♠❡♥t❛✐s✳ ❚❛♠❜é♠✱ ♦❜t❡♠♦s ❛s ❢ór♠✉❧❛s ❞❡ ❋r❡✉❞❡♥t❤❛❧ ❡ ❞❡ ❲❡②❧✱ s❡♥❞♦ q✉❡ ❛ ú❧t✐♠❛✱ ❝❛❧❝✉❧❛ ❛ ❞✐♠❡♥sã♦ ❞❛s r❡♣r❡s❡♥t❛çõ❡s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ Γλ ❡ ❡stá ❞❡s❝r✐t❛ ❛❜❛✐①♦✿

❞✐♠ Γλ = Y α∈R+

(λ+ρ, α) (ρ, α) ·

❋✐♥❛❧✐③❛♥❞♦✱ ❛ ❙❡çã♦ ✾✱ ❝❛r❛❝t❡r✐❛ ❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ❡①❝❡♣❝✐♦♥❛❧ g2 ❞❡

❞✐♠❡♥sã♦ ✶✹✳ ❯s❛♠♦s ♦ ❧✐✈r♦ ❘❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ❞♦s ❛✉t♦r❡s ❲✐❧❧✐❛♠ ❋✉❧t♦♥ ❡ ❏♦❡ ❍❛rr✐s ♣❛r❛ ❛ ❡❧❛❜♦r❛çã♦ ❞❡st❛ s❡çã♦ ❡ ❞❛ ❛♥t❡r✐♦r✳

(8)

❙❯▼➪❘■❖ ✈✐✐✐

❙✉♠ár✐♦

✶ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ✶

✶✳✶ ➪❧❣❡❜r❛s ❞❡ ▲✐❡✱ s✉❜á❧❣❡❜r❛s ❡ ✐❞❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❉❡r✐✈❛çõ❡s✱ ❤♦♠♦♠♦r✜s♠♦s✱ r❡♣r❡s❡♥t❛çõ❡s ❡ ♠ó❞✉❧♦s ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸ ➪❧❣❡❜r❛s ❞❡ ▲✐❡ s♦❧ú✈❡✐s ❡ ♥✐❧♣♦t❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✹ ❚❡♦r❡♠❛s ❞❡ ❊♥❣❡❧ ❡ ▲✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✷ ➪❧❣❡❜r❛s ❞❡ ▲✐❡ ❙❡♠✐ss✐♠♣❧❡s ✶✸

✷✳✶ ❈r✐tér✐♦ ❞❡ s❡♠✐ss✐♠♣❧✐❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ❚❡♦r❡♠❛ ❞❡ ❲❡②❧✱ ❡❧❡♠❡♥t♦s s❡♠✐ss✐♠♣❧❡s ❡ ♥✐❧♣♦t❡♥t❡s ✳ ✳ ✳ ✳ ✶✺

✸ ❙✉❜á❧❣❡❜r❛s ❞❡ ❈❛rt❛♥ ✶✽

✸✳✶ ❉❡✜♥✐çã♦ ❞❛s s✉❜á❧❣❡❜r❛s ❞❡ ❈❛rt❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✷ ❊①✐stê♥❝✐❛ ❞❛s s✉❜á❧❣❡❜r❛s ❞❡ ❈❛rt❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✸ g s❡♠✐ss✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✹ ❘❡♣r❡s❡♥t❛çõ❡s ❞❡ sl(2,C) ✷✹

✹✳✶ sl2✲♠ó❞✉❧♦s✱ ♣❡s♦s ❡ ❡❧❡♠❡♥t♦s ♣r✐♠✐t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✹✳✷ ❘❡♣r❡s❡♥t❛çõ❡s ✐rr❡❞✉tí✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✺ ❙✐st❡♠❛ ❞❡ r❛í③❡s ✸✵

✺✳✶ ❙✐♠❡tr✐❛s ❡ ❙✐st❡♠❛s ❞❡ ❘❛í③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✺✳✷ ❋♦r♠❛s q✉❛❞rát✐❝❛s ✐♥✈❛r✐❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✺✳✸ P♦s✐çã♦ r❡❧❛t✐✈❛ ❡♥tr❡ ❞✉❛s r❛í③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✺✳✹ ❇❛s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✺✳✺ ▼❛tr✐③ ❞❡ ❈❛rt❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✺✳✻ ❙✐st❡♠❛s ❞❡ r❛í③❡s ✐rr❡❞✉tí✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✺✳✼ ❉✐❛❣r❛♠❛s ❞❡ ❉②♥❦✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✻ ❊str✉t✉r❛ ❞❛s ➪❧❣❡❜r❛s ❞❡ ▲✐❡ s❡♠✐ss✐♠♣❧❡s ✹✵

(9)

❙❯▼➪❘■❖ ✐①

✻✳✷ ❙✉❜á❧❣❡❜r❛s ❞❡ ❇♦r❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✼ ❘❡♣r❡s❡♥t❛çõ❡s ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡ s❡♠✐ss✐♠♣❧❡s ✹✾

✼✳✶ ❊♥✈♦❧✈❡♥t❡ ✉♥✐✈❡rs❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✼✳✷ P❡s♦s ❡ ❡❧❡♠❡♥t♦s ♣r✐♠✐t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✼✳✸ ▼ó❞✉❧♦s ✐rr❡❞✉tí✈❡✐s ❝♦♠ ♣❡s♦ ♠❛✐s ❛❧t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✼✳✹ ▼ó❞✉❧♦s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✽ ❋ór♠✉❧❛s ❞❡ ❋r❡✉❞❡♥t❤❛❧ ❡ ❲❡②❧ ✻✶

✽✳✶ ❈â♠❛r❛s ❞❡ ❲❡②❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✽✳✷ ❈❛r❛❝t❡r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✽✳✸ ❖ ❛♥❡❧ ❞❡ r❡♣r❡s❡♥t❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✽✳✹ ❋ór♠✉❧❛ ❞❡ ❋r❡✉❞❡♥t❤❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✽✳✺ ❋ór♠✉❧❛s ❞❡ ❲❡②❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽

✾ ❆ á❧❣❡❜r❛ ❡①❝❡♣❝✐♦♥❛❧ g2 ✽✺

✾✳✶ ●r✉♣♦ ❞❡ ❲❡②❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✾✳✷ ❈♦♥str✉çã♦ ❞❡g2 ♣❡❧♦ s❡✉ ❞✐❛❣r❛♠❛ ❞❡ ❉②♥❦✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻

(10)

✶

✶ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s

❊st❛ s❡çã♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❞❡✜♥✐r ♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❛s á❧❣❡❜r❛s ❞❡ ▲✐❡ q✉❡ s❡rã♦ ❝♦♥st❛♥t❡♠❡♥t❡ ✉s❛❞♦s ♥❡st❡ tr❛❜❛❧❤♦✳

✶✳✶ ➪❧❣❡❜r❛s ❞❡ ▲✐❡✱ s✉❜á❧❣❡❜r❛s ❡ ✐❞❡❛✐s

❉❡✜♥✐çã♦ ✶✳✶✳✶✳ ❙❡Aé ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡K ❥✉♥t❛♠❡♥t❡ ❝♦♠ ✉♠❛ ❧❡✐

❞❡ ❝♦♠♣♦s✐çã♦ ❜✐❧✐♥❡❛r A×A→A ✭♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❛ss♦❝✐❛t✐✈❛✮✱ ❡♥tã♦

❞✐③❡♠♦s q✉❡ A é ✉♠❛ K✲á❧❣❡❜r❛✳ ❖✉ s❡❥❛✱ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ K✲❧✐♥❡❛r A⊗K A→A✳

❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❯♠❛ K✲á❧❣❡❜r❛ g ❝♦♠ ❛ ♦♣❡r❛çã♦ [ , ] : g×g → g é

❝❤❛♠❛❞❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ s❡✿

(L1) [X, X] = 0, ∀X ∈g✱ ✐st♦ é✱[ , ] é ❛❧t❡r♥❛❞❛✳ (L2) ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❏❛❝♦❜✐ é s❛t✐s❢❡✐t❛✿

[X,[Y, Z]] + [Y,[Z, X]] + [Z,[X, Y]] = 0, ∀X, Y, Z ∈g.

❆ ❞✐♠❡♥sã♦ ❞❛ á❧❣❡❜r❛ ❞❡ ▲✐❡g é ❛ ❞✐♠❡♥sã♦ ❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s✉❜❥❛❝❡♥t❡

❡✱ ❣❡r❛❧♠❡♥t❡✱ ❝❤❛♠❛♠♦s [ , ] ❞❡ ❝♦♠✉t❛❞♦r ♦✉ ❝♦❧❝❤❡t❡✳

❖❜s❡r✈❛çã♦ ✶✳✶✳✶✳ P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞❛ á❧❣❡❜r❛ ❡①t❡r✐♦r✱ ✉♠❛ á❧✲ ❣❡❜r❛ ❞❡ ▲✐❡ g ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ q✉❡ ♦ ♠❛♣❛ g⊗K g → g s❡ ❢❛t♦r❛

❡♠

g⊗K g→

^2

g→g.

◆♦t❡ q✉❡ ❛♣❧✐❝❛♥❞♦ ♦ ✐t❡♠ (L1) ❛♦ ❡❧❡♠❡♥t♦X+Y✱ ♦❜t❡♠♦s [X, Y] =

−[Y, X] ♣❛r❛ t♦❞♦X, Y ∈g ✭❛♥t✐✲s✐♠❡tr✐❛✮ ❡ ❝❛s♦ ❝❤❛r K 6= 2✱ ❛ r❡❝í♣r♦❝❛ é

✈á❧✐❞❛✱ ✐st♦ é✱ [X, Y] =−[Y, X]⇒[X, X] = 0 ♣❛r❛ X, Y ∈g✳

❊①❡♠♣❧♦ ✶✳✶✳✶✳ ◗✉❛♥❞♦Aé ✉♠❛K✲á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡rá✲

❧❛ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ❞❡✜♥✐♥❞♦ ♦ ❝♦♠✉t❛❞♦r [X, Y] =X ∗Y −Y ∗X ♣❛r❛ X, Y ∈ A✱ ♦♥❞❡ ∗ é ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠ A✳ ❆ss✐♠✱ ❛ ❛ss♦❝✐❛çã♦ A 7→ AL

✭AL é ❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ✐♥❞✉③✐❞❛ ♣♦r A✮ ❞❡✜♥❡ ✉♠ ❢✉♥t♦r ❞❛ ❝❛t❡❣♦r✐❛ ❞❛s K✲á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s ♣❛r❛ ❛ ❝❛t❡❣♦r✐❛ ❞❛s á❧❣❡❜r❛s ❞❡ ▲✐❡ s♦❜r❡K✱ ♣♦✐s

(i) s❡f :A→B ✉♠ K✲❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s✱ t❡♠✲s❡

(11)

✶✳✶ ➪❧❣❡❜r❛s ❞❡ ▲✐❡✱ s✉❜á❧❣❡❜r❛s ❡ ✐❞❡❛✐s ✷

■st♦ ♠♦str❛ q✉❡ f ✐♥❞✉③ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡ fL:AL →BL✳

(ii) ❝♦♥s✐❞❡r❡ ♦s K✲❤♦♠♦♠♦r✜s♠♦s ❞❡ á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s A −→f B −→g C ❡

♦s ✐♥❞✉③✐❞♦s AL fL

−→BL gL

−→CL✳ P♦❞❡✲s❡ ✈❡r✐✜❝❛r q✉❡(g◦f)L =gL◦fL✳ (iii) ❛ ✐❞❡♥t✐❞❛❞❡ ❡♠ A ✐♥❞✉③ ❛ ✐❞❡♥t✐❞❛❞❡ ❡♠ AL✳

❆ ❢✉♥t♦r✐❛❧✐❞❛❞❡ ❝✐t❛❞❛ ❛❝✐♠❛ é ✐♠♣♦rt❛♥t❡ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ s❡ A∼=B

✭❡♥tr❡ á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s✮✱ ❡♥tã♦ AL∼=BL ✭❡♥tr❡ á❧❣❡❜r❛s ❞❡ ▲✐❡✮✳

❊①❡♠♣❧♦ ✶✳✶✳✷✳ ❙❡❥❛V é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ns♦❜r❡ ✉♠ ❝♦r♣♦ K✳ ❉❡♥♦t❡♠♦s ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡♥❞♦♠♦r✜s♠♦s ❞❡V ♣♦r

❊♥❞(V) ={ϕ :V →V | ϕ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r}✳

❆ss✐♠✱ ❊♥❞(V)é ✉♠❛K✲á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❡ s❡ ✜①❛r♠♦s ✉♠❛ ❜❛s❡ ♣❛r❛V✱

❊♥❞(V) ∼= Mn×n(K)✱ ♦♥❞❡ Mn×n(K) é ❛ á❧❣❡❜r❛ ✭❛ss♦❝✐❛t✐✈❛✮ ❞❛s ♠❛tr✐③❡s

n×n ❝♦♠ ❡♥tr❛❞❛s ❡♠K✳ P❡❧♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ❊♥❞(V) é ✉♠❛ á❧❣❡❜r❛ ❞❡

▲✐❡ ❡ t❡♠ ❞✐♠❡♥sã♦ n2✳ ❯s❛r❡♠♦s ❛ ♥♦t❛çã♦ gl(V) ♦✉ gl(n, K) q✉❛♥❞♦ ♥♦s

r❡❢❡r✐r♠♦s ❛ ❊♥❞(V) ❝♦♠♦ á❧❣❡❜r❛ ❞❡ ▲✐❡✱ ❝❤❛♠❛❞❛ á❧❣❡❜r❛ ❣❡r❛❧ ❧✐♥❡❛r✳

❊①❡♠♣❧♦ ✶✳✶✳✸✳ ❖❜s❡r✈❡ q✉❡ s❡ V é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ n

s♦❜r❡ ✉♠ ❝♦r♣♦ K✱ ♣♦❞❡♠♦s ✈❡r V ❝♦♠♦ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ❞❡✜♥✐♥❞♦ [X, Y] = 0♣❛r❛ t♦❞♦X, Y ∈V✳ ❆s á❧❣❡❜r❛s ❞❡ ▲✐❡ ❞❡st❛ ❢♦r♠❛ sã♦ ❝❤❛♠❛❞❛s

❛❜❡❧✐❛♥❛s✳

❉❡✜♥✐çã♦ ✶✳✶✳✸✳ ❯♠ s✉❜❡s♣❛ç♦ h ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡gé ❞✐t♦ s✉❜á❧❣❡✲

❜r❛ ❞❡ ▲✐❡ ✭♦✉ ❛♣❡♥❛s s✉❜á❧❣❡❜r❛✮ s❡ [X, Y]∈h ♣❛r❛ t♦❞♦X, Y ∈h✳

❊①❡♠♣❧♦ ✶✳✶✳✹✳ ❈♦♥s✐❞❡r❡ B ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡g✱ ❡♥tã♦ ♦ ❝❡♥tr❛❧✐③❛❞♦r

❞❡ B ❡♠ g é ✉♠❛ s✉❜á❧❣❡❜r❛ ❞❡ g ✭♣❡❧❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❏❛❝♦❜✐✮ ❞❡✜♥✐❞❛ ♣♦r Cg(B) ={X ∈g | [X, Y] = 0,∀Y ∈B}✳

❉❡✜♥✐çã♦ ✶✳✶✳✹✳ ❯♠❛ s✉❜á❧❣❡❜r❛ a ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ g é ❝❤❛♠❛❞❛

✐❞❡❛❧ s❡ ∀X ∈ a ❡ ∀Y ∈ g t❡♠✲s❡ [X, Y] ∈ a✳ ❈♦♠♦ [X, Y] = −[Y, X]

♣❛r❛ t♦❞♦ X, Y ∈ g✱ ❛s ♥♦çõ❡s ❞❡ ✐❞❡❛✐s à ❡sq✉❡r❞❛ ❡ ❞✐r❡✐t❛ ❝♦✐♥❝✐❞❡♠✱

❞✐❢❡r❡♥t❡♠❡♥t❡ ❞❛ t❡♦r✐❛ ❞❡ ❛♥é✐s✳

(12)

✶✳✶ ➪❧❣❡❜r❛s ❞❡ ▲✐❡✱ s✉❜á❧❣❡❜r❛s ❡ ✐❞❡❛✐s ✸

❊①❡♠♣❧♦ ✶✳✶✳✻✳ ❉❡♥♦t❡ sl(V) ={φ ∈ gl(V) | tr(φ) = 0}✳ ❆✜r♠❛♠♦s q✉❡✱

sl(V) é ✐❞❡❛❧ ❞❡ gl(V)✳ ❉❡ ❢❛t♦✱ ✉s❛♥❞♦ ❛ ✐❞❡♥t✐✜❝❛çã♦ ❊♥❞(V)∼=Mn×n(K)✱

s❛❜❡♠♦s q✉❡ ❛ ❛♣❧✐❝❛çã♦ tr❛ç♦ é ❧✐♥❡❛r ❡ q✉❡ tr(M N) = tr(N M) ♣❛r❛ t♦❞♦ M, N ∈Mn×n(K)✳ ▲♦❣♦✱ s❡ M, N ∈sl(V)❡ λ∈K ♦❜t❡♠♦s✿

(i) tr(M+N) = tr(M) +tr(N) = 0 ❡ tr(λM) = λ tr(M) = 0✱ ✐st♦ é✱ sl(V)é

s✉❜❡s♣❛ç♦ ❞❡ gl(V)✳

(ii) tr([M, N]) =tr(M N−N M) = tr(M N)−tr(N M) = 0✳

❆ss✐♠✱ (i)❡(ii)♠♦str❛♠ q✉❡sl(V)é ✐❞❡❛❧ ❞❡ gl(V)✳ ❈❤❛♠❛♠♦ssl(V)

❞❡ á❧❣❡❜r❛ ❡s♣❡❝✐❛❧ ❧✐♥❡❛r✳

❖❜s❡r✈❛çã♦ ✶✳✶✳✷✳ ❆s s✉❜á❧❣❡❜r❛s ❞❡gl(V)sã♦ ❝❤❛♠❛❞❛s á❧❣❡❜r❛s ❞❡ ▲✐❡

❧✐♥❡❛r❡s✳

❉❡✜♥✐çã♦ ✶✳✶✳✺✳ ❉✐③❡♠♦s q✉❡ g é s✐♠♣❧❡s s❡ ❞✐♠ g > 1 ❡ g ♥ã♦ ♣♦ss✉✐

✐❞❡✐❛s ❞✐❢❡r❡♥t❡s ❞❡ {0} ❡ g✳

❊①❡♠♣❧♦ ✶✳✶✳✼✳ ◗✉❛♥❞♦ ❝❤❛r K 6= 2✱ ♦ ✐❞❡❛❧ sl(2, K) ❞❡ gl(2, K) ❞❛s ♠❛✲

tr✐③❡s 2×2 ❞❡ tr❛ç♦ ♥✉❧♦ ❝♦♠ ❡♥tr❛❞❛s ❡♠ K é s✐♠♣❧❡s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡

❝❤❛r K = 0✱ ❡♥tã♦ sl(n, K) é s✐♠♣❧❡s✳

❊①❡♠♣❧♦ ✶✳✶✳✽✳ ❖ ❝❡♥tr♦ ❞❡ g✱ ❞❡✜♥✐❞♦ ♣♦r

z(g) ={X ∈g | [X, Y] = 0,∀Y ∈g},

é ✉♠ ✐❞❡❛❧ ❞❡ g✳ ◆♦t❡ q✉❡g é ❛❜❡❧✐❛♥♦ s❡✱ ❡ só s❡✱ z(g) =g✳

❊①❡♠♣❧♦ ✶✳✶✳✾✳ ❙❡❥❛ h s✉❜á❧❣❡❜r❛ ❞❡ g✳ ❖ ♥♦r♠❛❧✐③❛❞♦r ❞❡ h ❡♠ g é

❞❡✜♥✐❞♦ ♣♦r ng(h) = {X ∈ g | [X, Y] ∈ h,∀Y ∈ h}✳ ❉❡st❡ ♠♦❞♦✱ ng(h) é ❛ ♠❛✐♦r s✉❜á❧❣❡❜r❛ ❞❡ g ♦♥❞❡ h é ✐❞❡❛❧✳

❊①❡♠♣❧♦ ✶✳✶✳✶✵✳ ❆ á❧❣❡❜r❛ ❞❡r✐✈❛❞❛ ❞❡ g✱ ❞❡♥♦t❛❞❛ [g,g]✱ é ❛ á❧❣❡❜r❛

❣❡r❛❞❛ ♣♦r t♦❞♦s ♦s ❝♦♠✉t❛❞♦r❡s [X, Y] ❝♦♠ X, Y ∈ g✳ P♦rt❛♥t♦✱ [g,g] =

{P[Xi, Yj] | Xi, Yj ∈g}✳ ➱ ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡[g,g] é ✐❞❡❛❧ ❞❡g ❡ q✉❛♥❞♦ gé

s✐♠♣❧❡s✱ z(g) = 0 ❡[g,g] =g✳

❙❡ a é ✉♠ ✐❞❡❛❧ ❞❡ g✱ ❞❡✜♥✐♠♦s ❛ á❧❣❡❜r❛ q✉♦❝✐❡♥t❡ ❝♦♠♦ ♦ ❡s♣❛ç♦ g/a={X+a | X ∈g}✳ ❆ss✐♠✱ g/at♦r♥❛✲s❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ✈✐❛

(13)

✶✳✷ ❉❡r✐✈❛çõ❡s✱ ❤♦♠♦♠♦r✜s♠♦s✱ r❡♣r❡s❡♥t❛çõ❡s ❡ ♠ó❞✉❧♦s ✹

✶✳✷ ❉❡r✐✈❛çõ❡s✱ ❤♦♠♦♠♦r✜s♠♦s✱ r❡♣r❡s❡♥t❛çõ❡s ❡ ♠ó✲

❞✉❧♦s

❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❙❡❥❛ A ✉♠❛ K✲á❧❣❡❜r❛✳ ❯♠❛ ❞❡r✐✈❛çã♦ ❞❡ A é ✉♠❛

❛♣❧✐❝❛çã♦ K✲❧✐♥❡❛rδ :A→A s❛t✐s❢❛③❡♥❞♦ δ(ab) =aδ(b) +δ(a)b, ∀a, b∈A✳

■r❡♠♦s ❞❡♥♦t❛r ♦ ❝♦♥❥✉♥t♦ ❞❛s ❞❡r✐✈❛çõ❡s ❞❡ A ♣♦r ❉❡r(A)✱ q✉❡ é s✉✲

❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❊♥❞(A)✳ ❉❡st❛❝❛♠♦s t❛♠❜é♠ q✉❡ ♦ ❝♦♠✉t❛❞♦r ❞❡ ❞✉❛s

❞❡r✐✈❛çõ❡s é ✉♠❛ ❞❡r✐✈❛çã♦✱ ♠❛s ❛ ❝♦♠♣♦s✐çã♦ ♣♦❞❡ ♥ã♦ s❡r✳ ❉✐❛♥t❡ ❞✐ss♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ s❡gé ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✱ ❡♥tã♦ ❉❡r(g)é ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡

✭s✉❜á❧❣❡❜r❛ ❞❡ gl(g)✮ ✈✐❛

[D1, D2] =D1◦D2−D2◦D1 ♣❛r❛ t♦❞♦D1, D2 ∈ ❉❡r(g).

❊①❡♠♣❧♦ ✶✳✷✳✶✳ ❖ ♣r✐♥❝✐♣❛❧ ❡①❡♠♣❧♦ ❞❡ ❞❡r✐✈❛çã♦ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ é ❛ ❛♣❧✐❝❛çã♦ ❛❞❥✉♥t❛ ❛❞(X) :g→g❞❛❞❛ ♣♦r ❛❞(X)Y = [X, Y]✱ ♦♥❞❡X ∈g

é ✜①♦✳ P❛r❛ ✈❡r✐✜❝❛r ❡st❡ ❢❛t♦✱ ❜❛st❛ ✉s❛r ❛ ❜✐❧✐♥❡❛r✐❞❛❞❡ ❞♦ ❝♦♠✉t❛❞♦r ❡ ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❏❛❝♦❜✐ ✭❝♦♥s✐❞❡r❡ X, Y, Z ∈g ❡ α∈K✮✿

(i)❛❞(X)(αY +Z) = [X, αY +Z] =α[X, Y] + [X, Z] =α❛❞(X)Y +❛❞(X)Z✳ (ii) ❛❞(X)([Y, Z]) = [X,[Y, Z]] = −[Y,[Z, X]]−[Z,[X, Y]] = [Y,[X, Z]] + [[X, Y], Z]] = [Y,❛❞(X)Z] + [❛❞(X)Y, Z]✳

❆s ❞❡r✐✈❛çõ❡s ❞❡st❛ ❢♦r♠❛ sã♦ ❝❤❛♠❛❞❛s ❞❡r✐✈❛çõ❡s ✐♥t❡r♥❛s ❡ ❛s ❞❡♠❛✐s ❡①t❡r♥❛s✳

❉❡✜♥✐çã♦ ✶✳✷✳✷✳ ❙❡❥❛♠ g1 ❡ g2 ❞✉❛s á❧❣❡❜r❛s ❞❡ ▲✐❡✳ ❯♠❛ tr❛♥s❢♦r♠❛çã♦

❧✐♥❡❛rϕ:g₁ →g₂ é ❞✐t❛ ❤♦♠♦♠♦r✜s♠♦ s❡ϕ([X, Y]) = [ϕ(X), ϕ(Y)]✳ ❈❛s♦ ϕ ❢♦r ❜✐❥❡t✐✈❛✱ ❞✐③❡♠♦s q✉❡ϕ é ✉♠ ✐s♦♠♦r✜s♠♦ ❡ q✉❡g1 ❡g2 sã♦ ✐s♦♠♦r❢❛s✳

❯♠ ❛✉t♦♠♦r✜s♠♦ ❞❡ g✱ ❞❡♥♦t❛♠♦s ♣♦r ❆✉t(g)✱ é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡g ❡♠ g✳

❖❜s❡r✈❛çã♦ ✶✳✷✳✶✳ ❖ ❑❡r✭ϕ✮ é ✐❞❡❛❧ ❞❡g1 ❡ ■♠✭ϕ✮ é s✉❜á❧❣❡❜r❛ ❞❡ g2✳

◆❡st❡ ❝♦♥t❡①t♦✱ sã♦ ✈á❧✐❞♦s ♦s três ❚❡♦r❡♠❛s ❞♦ ■s♦♠♦r✜s♠♦✿

❚❡♦r❡♠❛ ✶✳✷✳✶ ✭❚❡♦r❡♠❛s ❞♦ ■s♦♠♦r✜s♠♦✮✳ (a) ❙❡ ϕ :g1 →g2 é ✉♠ ❤♦✲

♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡✱ ❡♥tã♦ g₁/❑❡r(ϕ)∼=■♠(ϕ)✳

(b) ❙❡ a ❡ b sã♦ ✐❞❡❛✐s ❞❡ g✱ ❡♥tã♦ (a+b)/b ∼=a/(a∩b)✳

(14)

✶✳✷ ❉❡r✐✈❛çõ❡s✱ ❤♦♠♦♠♦r✜s♠♦s✱ r❡♣r❡s❡♥t❛çõ❡s ❡ ♠ó❞✉❧♦s ✺

❉❡♠♦♥str❛çã♦✳ ➱ ❛♥á❧♦❣❛ à ✈❡rsã♦ ♣❛r❛ ❣r✉♣♦s ❝♦♠ ❛s ❛❧t❡r❛çõ❡s ó❜✈✐❛s✳

❈♦♥s✐❞❡r❡ g ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ❝♦♠ ❜❛s❡{X1, ..., Xn}✳ ❈♦♠♦ [Xi, Xj]

é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ g✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r [Xi, Xj] = n

X

k=1

ak_ijXk✱ ♦♥❞❡ ak ij sã♦

❝❤❛♠❛❞❛s ❝♦♥st❛♥t❡s ❞❡ ❡str✉t✉r❛✳ ❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ ♠♦str❛ q✉❡ ❛s á❧❣❡❜r❛s ❞❡ ▲✐❡ sã♦ ❝♦♠♣❧❡t❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛s ♣❡❧❛s ❝♦♥st❛♥t❡s ❞❡ ❡str✉t✉r❛s ✭❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✮✳

▲❡♠❛ ✶✳✷✳✶✳ ❙❡❥❛♠ g1 ❡ g2 ❞✉❛s á❧❣❡❜r❛s ❞❡ ▲✐❡ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛✳ g1 é

✐s♦♠♦r❢❛ g₂ s❡✱ ❡ só s❡✱ ❡①✐st❡♠ ❜❛s❡s B1 ❡ B2 ❞❡ g1 ❡ g2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱

q✉❡ ♣♦ss✉❡♠ ❛s ♠❡s♠❛s ❝♦♥st❛♥t❡s ❞❡ ❡str✉t✉r❛✳

❉❡♠♦♥str❛çã♦✳ (⇒) ❈♦♠♦ g₁ ❡ g₂ sã♦ ✐s♦♠♦r❢❛s✱ ♣♦ss✉❡♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦

❞❡ ❡❧❡♠❡♥t♦s ♥❛ ❜❛s❡ ❡ ❡①✐st❡ ✉♠ ✐s♦♠♦r✜s♠♦ ϕ:g1 →g2 q✉❡ ❧❡✈❛ ✉♠❛ ❜❛s❡

❞❡ g1 ❡♠ ✉♠❛ ❜❛s❡ ❞❡ g2✳ ❚♦♠❡ {X1, ..., Xn} ❡{Y1, ..., Yn} ❜❛s❡s ❞❡g1 ❡ g2✱

r❡s♣❡❝t✐✈❛♠❡♥t❡✱ t❛✐s q✉❡ ϕ(Xi) = Yi✳ ❆ss✐♠✱

[Xi, Xj] = n

X

k=1

ak_ijXk ❡ [Yi, Yj] = n

X

k=1

bk_ijYk✳

❉❡st❡ ♠♦❞♦✱

[Yi, Yj] = [ϕ(Xi), ϕ(Xj)] =ϕ([Xi, Xj]) =

= ϕ n

X

k=1

akijXk

!

= n

X

k=1

akijϕ(Xk) = n

X

k=1

akijYk.

▲♦❣♦✱ Pn

k=1bkijYk =

Pn

k=1akijYk✳ ❙❛❜❡♠♦s q✉❡ {Y1, ..., Yn} é ❜❛s❡✱ ♣♦rt❛♥t♦✱

bk

ij =akij ♣❛r❛ t♦❞♦ k= 1, ..., n.

(⇐)❈♦♠♦g₁ ❡g₂ ♣♦ss✉❡♠ ❛s ♠❡s♠❛s ❝♦♥st❛♥t❡s ❞❡ ❡str✉t✉r❛✱ ❛s ❞✐♠❡♥sõ❡s

❞❡ g1 ❡ g2 sã♦ ✐❣✉❛✐s✳ ❙❡♥❞♦ ❛ss✐♠✱ ❝♦♥s✐❞❡r❡ {X1, ..., Xn} ❜❛s❡ ❞❡ g1 ❡

{Y1, ..., Yn}❜❛s❡ ❞❡ g2✳ ❉❡✜♥❛ ϕ:g1 →g2 t❛❧ q✉❡ ϕ(Xi) =Yi ❡ ❡st❡♥❞❛ ♣♦r

❧✐♥❡❛r✐❞❛❞❡ ♣❛r❛ ♦❜t❡r♠♦s ϕ([X, Y]) = [ϕ(X), ϕ(Y)], ∀X, Y ∈g₁✳

❉❡✜♥✐çã♦ ✶✳✷✳✸✳ ❯♠❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ g s♦❜r❡ K

é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ρ : g → gl(V)✱ ♦♥❞❡ V é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ✭s♦❜r❡ ♦

♠❡s♠♦ ❝♦r♣♦✮ ❡ ❞✐③❡♠♦s q✉❡ ❛ ❞✐♠❡♥sã♦ ❞❛ r❡♣r❡s❡♥t❛çã♦ é ❛ ❞✐♠❡♥sã♦ ❞❡

V✳

❊①❡♠♣❧♦ ✶✳✷✳✷✳ ❱✐♠♦s q✉❡ ❛❞(X) ∈ ❉❡r(g) ⊂ gl(g)✳ ❉❡st❡ ♠♦❞♦✱ ❛ ❛♣❧✐✲

(15)

✶✳✷ ❉❡r✐✈❛çõ❡s✱ ❤♦♠♦♠♦r✜s♠♦s✱ r❡♣r❡s❡♥t❛çõ❡s ❡ ♠ó❞✉❧♦s ✻

✐❞❡♥t✐❞❛❞❡ ❞❡ ❏❛❝♦❜✐✮✱ ❝❤❛♠❛❞❛ r❡♣r❡s❡♥t❛çã♦ ❛❞❥✉♥t❛✳ ❱❡r✐✜❝❛✲s❡ ❞❡ ❢♦r♠❛ ✐♠❡❞✐❛t❛ q✉❡ ❑❡r✭❛❞✮=z(g)✳

❉❡✜♥✐çã♦ ✶✳✷✳✹✳ ❙❡❥❛g✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ s♦❜r❡K✳ ❯♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧V✱

s♦❜r❡ ♦ ♠❡s♠♦ ❝♦r♣♦K✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ✉♠❛ ♦♣❡r❛çã♦g×V →V ❞❛❞❛ ♣♦r (X, v) 7→ Xv✱ é ❝❤❛♠❛❞♦ g✲♠ó❞✉❧♦ s❡ ❛s ❝♦♥❞✐çõ❡s ❛❜❛✐①♦ sã♦ s❛t✐s❢❡✐t❛s

♣❛r❛ t♦❞♦ X, Y ∈g❀u, v ∈V❀ λ, µ∈K✿

✭▼✶✮ (λX+µY)v =λ(Xv) +µ(Y v)✳

✭▼✷✮ X(λu+µv) =λ(Xu) +µ(Xv)✳

✭▼✸✮ [X, Y]v =X(Y v)−Y(Xv)✳

❖❜s❡r✈❛çã♦ ✶✳✷✳✷✳ ❙❡ ρ : g → gl(V) é ✉♠❛ r❡♣r❡s❡♥t❛çã♦✱ ❛ ♦♣❡r❛çã♦ (X, v)7→Xv =ρ(X)(v) ❢❛③ ❝♦♠ q✉❡ V s❡❥❛ ✉♠ g✲♠ó❞✉❧♦✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱

s❡ V é ✉♠ g✲♠ó❞✉❧♦✱ ❛ ❛♣❧✐❝❛çã♦ ρ :g → gl(V) ❞❡✜♥✐❞❛ ♣♦r ρ(X)(v) = Xv

é ✉♠❛ r❡♣r❡s❡♥t❛çã♦✳ ❖✉ s❡❥❛✱ ❛s ❞❡✜♥✐çõ❡s ❞❡ r❡♣r❡s❡♥t❛çõ❡s ❡ ♠ó❞✉❧♦s sã♦ ❡q✉✐✈❛❧❡♥t❡s✳

❉❡✜♥✐çã♦ ✶✳✷✳✺✳ ❯♠❛ r❡♣r❡s❡♥t❛çã♦ ✭♦✉ ♠ó❞✉❧♦✮ ρ : g → gl(V) é ❞✐t❛

✐rr❡❞✉tí✈❡❧ s❡ V é ♥ã♦✲♥✉❧♦ ❡ ♥ã♦ ♣♦ss✉✐ s✉❜❡s♣❛ç♦s g✲✐♥✈❛r✐❛♥t❡s ✭s✉❜♠ó✲

❞✉❧♦s✮ ♣ró♣r✐♦s ♥ã♦✲tr✐✈✐❛✐s✱ ♦✉ s❡❥❛✱ s❡W é s✉❜❡s♣❛ç♦ ❞❡V t❛❧ q✉❡ρ(X)W ⊂

W ♣❛r❛ t♦❞♦X ∈g✱ ❡♥tã♦W =V ♦✉W = 0✳ ❉✐③❡♠♦s q✉❡ ❛ r❡♣r❡s❡♥t❛çã♦ρ

✭♦✉ ♠ó❞✉❧♦V✮ é ❝♦♠♣❧❡t❛♠❡♥t❡ r❡❞✉tí✈❡❧ s❡ ρé s♦♠❛ ❞✐r❡t❛ ❞❡ r❡♣r❡s❡♥✲

t❛çõ❡s

✐rr❡❞✉tí✈❡✐s✱ ♦✉ s❡❥❛✱V s❡ ❞❡❝♦♠♣õ❡ ❝♦♠♦ s♦♠❛ ❞✐r❡t❛V =V1⊕· · ·⊕Vn♦♥❞❡

❝❛❞❛ Vi é ✐♥✈❛r✐❛♥t❡ ♣❡❧❛ r❡♣r❡s❡♥t❛çã♦ ❡ ❛ r❡str✐çã♦ ❞❡ ρ ❛Vi é ✐rr❡❞✉tí✈❡❧✳

❊①❡♠♣❧♦ ✶✳✷✳✸✳ ❙❡❥❛ V ✉♠ g✲♠ó❞✉❧♦✳ ❖ ❡s♣❛ç♦ ❞✉❛❧ ❞❡ V∗ _{♣♦ss✉✐ ✉♠❛}

❡str✉t✉r❛ ❞❡ g✲♠ó❞✉❧♦ ❞❡✜♥✐❞❛ ♣♦r

g×V∗ _→ _V∗

(X, f) 7→ Xf

t❛❧ q✉❡ (Xf)v =−f(Xv) ♣❛r❛ v ∈ V✳ ❉❡ ❢❛t♦✱ ❛s ❞✉❛s ♣r✐♠❡✐r❛s ❝♦♥❞✐çõ❡s

❞❛ ❞❡✜♥✐çã♦ ❞❡ ♠ó❞✉❧♦ sã♦ ✈❡r✐✜❝❛❞❛s ❢❛❝✐❧♠❡♥t❡✳ P❛r❛ (M3)✱

([X, Y]f)v = −f([X, Y].v)

(16)

✶✳✸ ➪❧❣❡❜r❛s ❞❡ ▲✐❡ s♦❧ú✈❡✐s ❡ ♥✐❧♣♦t❡♥t❡s ✼

❊①❡♠♣❧♦ ✶✳✷✳✹✳ ❙❡❥❛♠ V, W ❞♦✐s g✲♠ó❞✉❧♦s✳ ❊♥tã♦✱ ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ V ⊗W t❛♠❜é♠ ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ❞❡ g✲♠ó❞✉❧♦ ✈✐❛

g×V ⊗W → V ⊗W (X, v⊗w) 7→ X(v⊗w),

♦♥❞❡ X(v⊗w) = (Xv)⊗w+v⊗(Xw)✳

❙❡♥❞♦ ❛ss✐♠✱ ❝♦♥s✐❞❡r❡ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ❡ ❛

❛♣❧✐❝❛çã♦

ϕ : V∗_⊗_V _→ _❊♥❞_(V₎

f⊗v 7→ g

❞❡✜♥✐❞❛ ♣♦r g(w) = f(w)v ♣❛r❛ w ∈ V✳ ❯s❛♥❞♦ ❛ ❜❛s❡ ❞✉❛❧✱ ❝♦♥❝❧✉✐♠♦s

q✉❡ ϕ é s♦❜r❡❥❡t✐✈❛ ❡ ❝♦♠♦ ♦ ❞♦♠í♥✐♦ ❡ ❝♦♥tr❛❞♦♠í♥✐♦ ♣♦ss✉❡♠ ❛ ♠❡s♠❛

❞✐♠❡♥sã♦✱ t❡♠♦s ✉♠ ✐s♦♠♦r✜s♠♦✳ ❱✐♠♦s q✉❡ V∗_⊗_V _{♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦}

✉♠ g✲♠ó❞✉❧♦✱ ❧♦❣♦ ❊♥❞(V)t❛♠❜é♠ ♦ é✳

✶✳✸ ➪❧❣❡❜r❛s ❞❡ ▲✐❡ s♦❧ú✈❡✐s ❡ ♥✐❧♣♦t❡♥t❡s

❉❡✜♥✐çã♦ ✶✳✸✳✶✳ ❙❡❥❛ g✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✳ ❆ sér✐❡ ❞❡r✐✈❛❞❛ ❞❡ g é ✉♠❛

sér✐❡ ❞❡s❝❡♥❞❡♥t❡ ❞❡ ✐❞❡✐❛s ❞❡ g ❞❡✜♥✐❞❛ ♣♦r✿

g(0) =g ❡ g(n)= [g(n−1)_,_g(n−1)_] _♣❛r❛ _n_>₁_✳

❖❜s❡r✈❡ q✉❡ g(1) ⊇ · · · ⊇ g(i) ⊇ g(i+1) ⊇ · · ·✳ ❉✐③❡♠♦s q✉❡ g é s♦❧ú✈❡❧ s❡

❡①✐st❡ ✉♠ ✐♥t❡✐r♦ k t❛❧ q✉❡ g(k)= 0✳

Pr♦♣♦s✐çã♦ ✶✳✸✳✶✳ ❙❡❥❛ g ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✳

(a) ❙❡ g é s♦❧ú✈❡❧✱ ❡♥tã♦ t♦❞❛s ❛s s✉❛s s✉❜á❧❣❡❜r❛s ❡ ✐♠❛❣❡♥s ❤♦♠♦♠♦r❢❛s

❞❡ g t❛♠❜é♠ ♦ sã♦✳

(b) ❙❡ a é ✉♠ ✐❞❡❛❧ s♦❧ú✈❡❧ ❞❡ g t❛❧ q✉❡ g/a é s♦❧ú✈❡❧✱ ❡♥tã♦ g t❛♠❜é♠ é

s♦❧ú✈❡❧✳

(17)

✶✳✸ ➪❧❣❡❜r❛s ❞❡ ▲✐❡ s♦❧ú✈❡✐s ❡ ♥✐❧♣♦t❡♥t❡s ✽

❉❡♠♦♥str❛çã♦✳ (a)✿ ❙❡ a é s✉❜á❧❣❡❜r❛ ❞❡ g✱ ❡♥tã♦ a(i) ⊂ g(i) ♣❛r❛ t♦❞♦ i ❡

s❡ φ:g→b é ✉♠ ❤♦♠♦♠♦r✜s♠♦ s♦❜r❡❥❡t✐✈♦✱ ♣♦❞❡♠♦s ♠♦str❛r ♣♦r ✐♥❞✉çã♦

❡♠ iq✉❡ φ(g(i)_{) =} _b(i)_✳

(b)✿ P♦r ❤✐♣ót❡s❡✱ (g/a)(n) _{= 0} _{♣❛r❛ ❛❧❣✉♠ ✐♥t❡✐r♦} _n_{✳ ❚♦♠❡} _π _: _g _→ _g/a _❛

❛♣❧✐❝❛çã♦ ❝❛♥ô♥✐❝❛ ❞❡✜♥✐❞❛ ♣♦rπ(X) =X+a✳ P♦r (a)✱ π(g(i)_{) = (g/a)}(i)_⇒

π(g(n)_{) = 0}_⇒_g(n) _⊂_a_{✳ P♦r ♦✉tr♦ ❧❛❞♦✱}_a(m) _{= 0} _❡

g(n+m) = (g(n))(m)=⊂a(m) = 0.

(c)✿ P❡❧♦ ✐t❡♠(a)✱ a/(a∩b) é s♦❧ú✈❡❧ ❡ ♣♦r (b)✱a+b t❛♠❜é♠ ♦ é✳

▲❡♠❛ ✶✳✸✳✶✳ ❚♦❞❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ g❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ✐❞❡❛❧

♠❛①✐♠❛❧ s♦❧ú✈❡❧✳

❉❡♠♦♥str❛çã♦✳ ❙❛❜❡♠♦s q✉❡ ❛ s♦♠❛ ❞❡ t♦❞♦s ♦s ✐❞❡❛✐s s♦❧ú✈❡✐s ❞❡g ❛✐♥❞❛ é

✐❞❡❛❧ s♦❧ú✈❡❧ ❡ é ♠❛①✐♠❛❧✳ ❙✉♣♦♥❤❛ q✉❡ s1 ❡ s2 sã♦ ✐❞❡❛✐s s♦❧ú✈❡✐s ♠❛①✐♠❛✐s✱ ❡♥tã♦ s1+s2 ⊇s1 ❡s1+s2 ⊇s2✳ ❈♦♥s✐❞❡r❛♥❞♦ ❛ ♠❛①✐♠❛❧✐❞❛❞❡ ❞❡s1 ❡s2❡ ♦ ❢❛t♦ ❞❡ q✉❡ s♦♠❛ ❞❡ ✐❞❡❛✐s s♦❧ú✈❡✐s ❛✐♥❞❛ é s♦❧ú✈❡❧ t❡♠♦ss1=s1+s2=s2✳ ❉❡✜♥✐çã♦ ✶✳✸✳✷✳ ❖ ✐❞❡❛❧ s♦❧ú✈❡❧ ♠❛①✐♠❛❧ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡gé ❝❤❛♠❛❞♦

r❛❞✐❝❛❧ ❡ é ❞❡♥♦t❛❞♦ ♣♦r r(g)✳

❉❡✜♥✐çã♦ ✶✳✸✳✸✳ ❙❡ r(g) = 0✱ ❞✐③❡♠♦s q✉❡ ❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ g é s❡♠✐ss✐♠✲

♣❧❡s✳

❖❜s❡r✈❛çã♦ ✶✳✸✳✶✳ ◆❛ ❉❡✜♥✐çã♦ ✶✳✶✳✺✱ ❡①✐❣✐♠♦s q✉❡ ❞✐♠ g >1✳ P♦r ✐ss♦✱

❡①✐st❡ ✉♠❛ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ♥❛s ❞❡✜♥✐çõ❡s ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡ s✐♠♣❧❡s ❡ s❡♠✐s✲ s✐♠♣❧❡s ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ t♦❞❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ s✐♠♣❧❡s é s❡♠✐ss✐♠♣❧❡s✱ ♣♦✐s✱ ♣♦r ❞❡✜♥✐çã♦✱ á❧❣❡❜r❛s ❞❡ ▲✐❡ ✉♥✐❞✐♠❡♥s✐♦♥❛✐s ♥ã♦ sã♦ s❡♠✐ss✐♠♣❧❡s✳

◆♦t❡ q✉❡ gss =g/r(g) é s❡♠✐ss✐♠♣❧❡s✳ ❉❡ ❢❛t♦✱ t♦♠❡ I =I +r(g) ✉♠

✐❞❡❛❧ s♦❧ú✈❡❧ ❞❡gss✳ ▲♦❣♦✱I

(n)

= 0 ♣❛r❛ ❛❧❣✉♠ ✐♥t❡✐r♦ n✱ ✐st♦ é✱ I(n)_⊂_r(g)_❡

❝♦♠♦ r(g) é s♦❧ú✈❡❧ (I(n)₎(k) ₌_I(n+k) _{= 0}_{✳ ❉❛í✱} _I _⊂_r(g)_{❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡}

I = 0✳

❉❡✜♥✐çã♦ ✶✳✸✳✹✳ ❙❡❥❛ g✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✳ ❆ sér✐❡ ❝❡♥tr❛❧ ❞❡s❝❡♥❞❡♥t❡

❞❡ g é ✉♠❛ sér✐❡ ❞❡s❝❡♥❞❡♥t❡ ❞❡ ✐❞❡✐❛s ❞❡g ❞❡✜♥✐❞❛ ❞❛ ❢♦r♠❛✿

g1 =g ❡gn = [g,gn−1] ♣❛r❛ n>2✳

❉✐③❡♠♦s q✉❡ g é ♥✐❧♣♦t❡♥t❡ s❡ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦k t❛❧ q✉❡ gk= 0✳ ❚❛♠❜é♠

(18)

✶✳✸ ➪❧❣❡❜r❛s ❞❡ ▲✐❡ s♦❧ú✈❡✐s ❡ ♥✐❧♣♦t❡♥t❡s ✾

❊q✉✐✈❛❧❡♥t❡♠❡♥t❡✱gé ♥✐❧♣♦t❡♥t❡ s❡ ♣❛r❛ ❛❧❣✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦n ❡ ♣❛r❛

t♦❞♦ X0, . . . , Xn ∈g✱ t❡♠♦s

[X0,[X1,[. . . , Xn]. . .]] = (❛❞(X0)◦❛❞(X1)◦ · · · ◦❛❞(Xn−1))(Xn) = 0.

❊♠ ♣❛rt✐❝✉❧❛r✱ ❛❞(X)n−1 _{= 0} _{♣❛r❛ t♦❞♦} _X _∈ _g_{✳ ❖✉ s❡❥❛✱ s❡} _g _{é ♥✐❧♣♦✲}

t❡♥t❡✱ ❡♥tã♦ ❛❞(X)n _{= 0}_{♣❛r❛ t♦❞♦} _X _∈_g_{✳ ❱❡r❡♠♦s ❛❞✐❛♥t❡ q✉❡ ❛ r❡❝í♣r♦❝❛}

❞❡st❡ ❢❛t♦ é ✈❡r❞❛❞❡✐r❛ ✭r❡s✉❧t❛❞♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❚❡♦r❡♠❛ ❞❡ ❊♥❣❡❧✮✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦❜s❡r✈❡ q✉❡ t♦❞❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ♥✐❧♣♦t❡♥t❡ é s♦❧ú✈❡❧✱ ♣♦✐sg(i) ⊆gi

♣❛r❛ t♦❞♦ i ✐♥t❡✐r♦✳

Pr♦♣♦s✐çã♦ ✶✳✸✳✷✳ ❙❡❥❛ g ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✳

(a) ❙❡ g é ♥✐❧♣♦t❡♥t❡✱ ❡♥tã♦ t♦❞❛s ❛s s✉❛s s✉❜á❧❣❡❜r❛s ❡ ✐♠❛❣❡♥s ❤♦♠♦♠♦r✲

❢❛s ❞❡ g sã♦ ♥✐❧♣♦t❡♥t❡s✳

(b) ❙❡ g/z(g) é ♥✐❧♣♦t❡♥t❡✱ ❡♥tã♦ g t❛♠❜é♠ ♦ é✳ (c) ❙❡ g é ♥✐❧♣♦t❡♥t❡ ❡ ♥ã♦✲♥✉❧♦✱ ❡♥tã♦z(g)6= 0.

❉❡♠♦♥str❛çã♦✳ (a)✿ ❆♥á❧❛❣♦ ❛♦ ✐t❡♠ (a) ❞❛ Pr♦♣♦s✐çã♦ ✶✳✸✳✶✳

(b)✿ P♦r ❤✐♣ót❡s❡✱ (g/z(g))n _{= 0} _{♣❛r❛ ❛❧❣✉♠ ✐♥t❡✐r♦} _n_{✱ ❧♦❣♦} _gn _⊂ _z(g) _❡ gn+1 = [g,gn]⊂[g,z(g)] = 0✳

(c)✿ ❈♦♠♦gé ♥✐❧♣♦t❡♥t❡✱ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦n t❛❧ q✉❡gn= 0 ❡gn−1 6= 0✳ ❉❛í✱ gn = [g,gn−1] = 0 ♦ q✉❡ ✐♠♣❧✐❝❛ gn−1 ⊂z(g)✳

❊①❡♠♣❧♦ ✶✳✸✳✶✳ ❙❡ ❝❤❛r K = 2✱ ❡♥tã♦ g = sl(2, K) é ♥✐❧♣♦t❡♥t❡✳ ❈♦♠

❡❢❡✐t♦✱ ❝♦♥s✐❞❡r❡ ❛ ❜❛s❡ ❞❡ g✿

X =

0 1 0 0

✱Y =

0 0 1 0

✱ H =

1 0 0 −1

q✉❡ s❛t✐s❢❛③❡♠ ❛s r❡❧❛çõ❡s[X, Y] =H✱[H, X] = 2X = 0❡[H, Y] =−2Y = 0✳

▲♦❣♦✱

g2 = [g,g] =KH ❡g3 = [g,g2] = 0✳

❊①❡♠♣❧♦ ✶✳✸✳✷✳ ❆ á❧❣❡❜r❛ ❞❡ ▲✐❡2✲❞✐♠❡♥s✐♦♥❛❧g ❝♦♠ ❜❛s❡{X, Y}t❛❧ q✉❡

[X, Y] = X é s♦❧ú✈❡❧✱ ♠❛s ♥ã♦ ♥✐❧♣♦t❡♥t❡✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ gn = KX ♣❛r❛

t♦❞♦ n≥2✱ t❡♠♦s q✉❡ g ♥ã♦ é ♥✐❧♣♦t❡♥t❡✳ P♦r ♦✉tr♦ ❧❛❞♦✱

(19)

✶✳✹ ❚❡♦r❡♠❛s ❞❡ ❊♥❣❡❧ ❡ ▲✐❡ ✶✵

✶✳✹ ❚❡♦r❡♠❛s ❞❡ ❊♥❣❡❧ ❡ ▲✐❡

❱✐♠♦s ♥❛ ❙✉❜s❡çã♦ ✶✳✸ q✉❡ s❡ g é ♥✐❧♣♦t❡♥t❡✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ n t❛❧ q✉❡ ❛❞(X)n _{= 0} _{♣❛r❛ t♦❞♦} _X _∈ _g_{✳ ❆❣♦r❛✱ ✈❛♠♦s ♠♦str❛r ❛ r❡❝í♣r♦❝❛}

❞❡st❛ ❛✜r♠❛çã♦✱ ♠❛s ♣❛r❛ ✐ss♦✱ ✈❛♠♦s ✉t✐❧✐③❛r ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿ ▲❡♠❛ ✶✳✹✳✶✳ ❙❡ X ∈gl(V) é ♥✐❧♣♦t❡♥t❡✱ ❡♥tã♦ ❛❞(X) é ♥✐❧♣♦t❡♥t❡✳

❉❡♠♦♥str❛çã♦✳ ❙❛❜❡♠♦s q✉❡ ❛❞(X)Y =XY −Y X✳ ❉❡✜♥❛ ♦s ❡♥❞♦♠♦r✜s✲

♠♦s LX(Y) = XY ❡ RX(Y) = Y X t❛✐s q✉❡ ❛❞(X)Y = LX(Y)−RX(Y) = (LX −RX)(Y)✳ ◆♦t❡ q✉❡ LX ❡ RX ❝♦♠✉t❛♠✱ ♣♦✐s ❡♠ ❊♥❞✭❊♥❞(V))✱

LX(RX(Y)) = X(Y X) = (XY)X =RX(LX(Y))⇒LX ◦RX =RX ◦LX.

P♦r ❤✐♣ót❡s❡✱ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ n t❛❧ q✉❡ Xn _{= 0} _{❡ ❞❛í✱} _Ln

X(Y) = XnY = 0

❡ Rn

X(Y) = Y Xn= 0✳ ❈♦♠♦LX ❡ RX ❝♦♠✉t❛♠✱ ✈❛❧❡✿

(LX −RX)2n=

2n

X

i=0

2n i

L2n−i

X (−RX)i = 0.

P♦rt❛♥t♦✱ ❛❞(X) é ♥✐❧♣♦t❡♥t❡✳

❚❡♦r❡♠❛ ✶✳✹✳✶ ✭❊♥❣❡❧✮✳ ❙❡❥❛ g ✉♠❛ s✉❜á❧❣❡❜r❛ ❞❡ gl(V)✱ ♦♥❞❡ V é ✉♠ K✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛(♥ã♦✲♥✉❧♦)✳ ❙❡ g ❝♦♥s✐st❡ ❞❡ ❡♥❞♦♠♦r✲

✜s♠♦s ♥✐❧♣♦t❡♥t❡s✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ✈❡t♦r ♥ã♦✲♥✉❧♦ v ∈V t❛❧ q✉❡ gv = 0✱ ✐st♦

é✱ Xv = 0 ♣❛r❛ t♦❞♦ X ∈g✳

❉❡♠♦♥str❛çã♦✳ ❱❡❥❛ ❬✷❪ ✭♣á❣✐♥❛ ✶✷✮✳

❆♥t❡s ❞❡ ❛♣r❡s❡♥t❛r♠♦s ❛❧❣✉♠❛s ❝♦♥s❡q✉ê♥❝✐❛s ❞♦ ❚❡♦r❡♠❛ ❞❡ ❊♥❣❡❧✱ ♣r❡❝✐s❛♠♦s ❞❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿ s❡❥❛ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦

✜♥✐t❛ n✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❜❛♥❞❡✐r❛ ❡♠ V é ✉♠❛ ❝❛❞❡✐❛ ❞❡ s✉❜❡s♣❛ç♦s 0 =V0 ⊂V1 ⊂ · · · ⊂Vn =V✱ ♦♥❞❡ ❞✐♠ Vi =i ❡ q✉❡ X ∈ ❊♥❞(V) ❡st❛❜✐❧✐③❛

❛ ❜❛♥❞❡✐r❛ s❡ XVi ⊂Vi ♣❛r❛ t♦❞♦ i✳

❈♦r♦❧ár✐♦ ✶✳✹✳✶✳ ❙♦❜ ❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛ ❞❡ ❊♥❣❡❧✱ ❡①✐st❡ ✉♠❛ ❜❛♥❞❡✐r❛

{Vi}❡♠V ✐♥✈❛r✐❛♥t❡ s♦❜g✿ XVi ⊂Vi−1, ∀i= 1, . . . , n❡∀X ∈g✳ ❊♠ ♦✉tr❛s

♣❛❧❛✈r❛s✱ ❡①✐st❡ ✉♠❛ ❜❛s❡ ❞❡ V t❛❧ q✉❡ t♦❞♦ X ∈ g é r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠❛

♠❛tr✐③ ❡str✐t❛♠❡♥t❡ tr✐❛♥❣✉❧❛r✳

❉❡♠♦♥str❛çã♦✳ ❱❡❥❛ ❬✷❪ ✭♣á❣✐♥❛ ✶✸✮✳

(20)

✶✳✹ ❚❡♦r❡♠❛s ❞❡ ❊♥❣❡❧ ❡ ▲✐❡ ✶✶

❉❡♠♦♥str❛çã♦✳ ❱❡❥❛ ❬✷❪ ✭♣á❣✐♥❛ ✶✷✮✳

▲❡♠❛ ✶✳✹✳✷✳ ❙❡❥❛ a ✉♠ ✐❞❡❛❧ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ g⊂gl(V) ❡ λ∈a∗ _✉♠

❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r✳ ❙❡

W ={v ∈V | Xv =λ(X)v,∀X ∈a}✱

❡♥tã♦ gW ⊂W✳

❉❡♠♦♥str❛çã♦✳ ❱❡❥❛ ❬✶❪ ✭♣á❣✐♥❛ ✶✷✼✮✳

❚❡♦r❡♠❛ ✶✳✹✳✷ ✭▲✐❡✮✳ ❙❡❥❛ K ✉♠ ❝♦r♣♦ ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❢❡❝❤❛❞♦ ❞❡ ❝❛r❛❝✲

t❡ríst✐❝❛ ③❡r♦ ❡ g ✉♠❛ s✉❜á❧❣❡❜r❛ s♦❧ú✈❡❧ ❞❡ gl(V)✱ ♦♥❞❡ V é ✉♠ K✲❡s♣❛ç♦

✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ✭♥ã♦✲♥✉❧♦✮✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠ ✈❡t♦r ♥ã♦✲♥✉❧♦

v ∈ V t❛❧ q✉❡ Xv = λ(X)v,∀X ∈ g ❝♦♠ λ ∈ g∗✱ ♦✉ s❡❥❛✱ ❡①✐st❡ ✉♠ ✈❡t♦r

♥ã♦✲♥✉❧♦ ❡♠ V q✉❡ é ❛✉t♦✈❡t♦r ❝♦♠✉♠ ❛ t♦❞♦s ♦s X ∈g✳

■❞❡✐❛ ❞❛ ❞❡♠♦♥str❛çã♦✳ ❆ ❞❡♠♦str❛çã♦ é ❢❡✐t❛ ♣♦r ✐♥❞✉çã♦ ♥❛ ❞✐♠❡♥sã♦ ❞❡

g✱ s❡♥❞♦ q✉❡ ♦ ❝❛s♦ ❞✐♠ g = 1 s❡❣✉❡ ❞❛s ❤✐♣ót❡s❡s ✐♠♣♦st❛s ❛ K✳ ❯s❛♥❞♦

q✉❡ g é s♦❧ú✈❡❧✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ✉♠ ✐❞❡❛❧ h ⊂ g ❞❡ ❝♦❞✐♠❡♥sã♦ 1✳ P♦r

❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ ♦ ❝♦♥❥✉♥t♦ W = {v ∈ V | Xv = λ(X)v,∀X ∈ h} é

♥ã♦✲♥✉❧♦ ❡ ♣❡❧♦ ▲❡♠❛ ✶✳✹✳✷✱ gW ⊂ W✳ ❈♦♠♦ g = h+αZ ✭Z ∈ g❭h✮ ❡ Z

♣♦ss✉✐ ✉♠ ❛✉t♦✈❡t♦r v0 ❝♦♠ ❛✉t♦✈❛❧♦r µ0✱ ♣♦❞❡♠♦s ❡st❡♥❞❡r λ ❛♦ ❢✉♥❝✐♦♥❛❧

e

λ(Y +αZ) =λ(Y) +αµ0 ❡♠ g∗✳

❈♦r♦❧ár✐♦ ✶✳✹✳✸✳ ◆❛s ♠❡s♠❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛ ❞❡ ▲✐❡✱ g ❡st❛❜✐❧✐③❛

❛❧❣✉♠❛ ❜❛♥❞❡✐r❛ ❡♠ V✱ ♦✉ s❡❥❛✱ ❡①✐st❡ ✉♠❛ ❜❛s❡ ❞❡ V t❛❧ q✉❡ t♦❞♦ X ∈g é

r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠❛ ♠❛tr✐③ tr✐❛♥❣✉❧❛r s✉♣❡r✐♦r✳

❉❡♠♦♥str❛çã♦✳ ❱❡❥❛ ❬✷❪ ✭♣á❣✐♥❛ ✶✻✮✳

Pr♦♣♦s✐çã♦ ✶✳✹✳✶✳ ❙❡❥❛ g⊂gl(V) ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ s♦❜r❡ C✱ ♦♥❞❡ g ❡ V

sã♦ ❛♠❜♦s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ♥ã♦✲♥✉❧❛✳ ❆ss✐♠✱ t♦❞❛ r❡♣r❡s❡♥t❛çã♦ ✐rr❡❞✉tí✈❡❧ ❞❡ g é ❞❛ ❢♦r♠❛ V =V0⊗L✱ ♦♥❞❡ V0 é ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ✐rr❡❞✉tí✈❡❧ ❞❡ gss (✐st♦ é✱ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞❡gq✉❡ é tr✐✈✐❛❧ ❡♠r(g))❡Lé ✉♠❛ r❡♣r❡s❡♥t❛çã♦ 1✲❞✐♠❡♥s✐♦♥❛❧ ❞❡ g✳

❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ r(g) é s♦❧ú✈❡❧✱ ♦ ❚❡♦r❡♠❛ ❞❡ ▲✐❡ ❣❛r❛♥t❡ q✉❡ ❡①✐st❡ λ ∈ r(g)∗ _{t❛❧ q✉❡} _W ₌ _{_v _∈ _V _| _Xv ₌ _λ(X)v, _∀_X _∈ _r(g)_{} 6}_{= 0}_{✳ P❡❧♦ ▲❡♠❛}

✶✳✹✳✷✱ gW ⊂W ♦ q✉❡ ✐♠♣❧✐❝❛ V =W✳ ❆✜r♠❛♠♦s q✉❡ s❡ X ∈r(g)∩[g,g]✱

(21)

✶✳✹ ❚❡♦r❡♠❛s ❞❡ ❊♥❣❡❧ ❡ ▲✐❡ ✶✷

• ❙❡ X ∈ r(g)✱ tr(X) = λ(X)✭❞✐♠ V✮✱ ♣♦✐s ♦ ❚❡♦r❡♠❛ ❞❡ ▲✐❡ ♠♦str❛

q✉❡ ❡①✐st❡ ✉♠❛ ❜❛s❡ ❞❡ V t❛❧ q✉❡ X é r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠❛ ♠❛tr✐③

tr✐❛♥❣✉❧❛r s✉♣❡r✐♦r ❡ Xv =λ(X)v ♣❛r❛ t♦❞♦ X ∈r(g)✳

• ❙❡ X ∈ [g,g]✱ X ♣♦ss✉✐ tr❛ç♦ ♥✉❧♦ ✭♣♦✐s ♦ tr❛ç♦ é ❧✐♥❡❛r ❡ ♦ tr❛ç♦ ❞♦

❝♦♠✉t❛❞♦r ❞❡ ❞✉❛s ♠❛tr✐③❡s é s❡♠♣r❡ ③❡r♦✮✳

P♦rt❛♥t♦✱ ❝❛s♦ X ∈ r(g) ∩[g,g]✱ ♦❜t❡♠♦s q✉❡ λ(X)✭❞✐♠ V = 0✮ ❡

❝♦♥s❡q✉❡♥t❡♠❡♥t❡ λ(X) = 0✳ ❆❣♦r❛✱ ❡st❡♥❞❛ λ ❛ ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r eλ∈g∗

t❛❧ q✉❡ e_λ([g,_{g]) = 0}_{✳ ❆ss✐♠✱} e_λ _{✐♥❞✉③ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡}

g/[g,g] → C=gl(1,C) X+ [g,g] 7→ eλ(X)

q✉❡ é ✉♠❛ r❡♣r❡s❡♥t❛çã♦1✲❞✐♠❡♥s✐♦♥❛❧ ❞❡g❡ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♣♦✐seλ([g,g]) = 0✳ ❙❡ ❝♦♥s✐❞❡r❛r♠♦sC=L✱ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ g❡♠ L é

g×L → L (X, v) 7→ eλ(X)v.

P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♥s✐❞❡r❡ ♦✉tr❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ g

g×(V ⊗_CL∗₎ _→ _V _⊗

CL∗

(X, v⊗ψ) 7→ Xv⊗ψ+v⊗Xψ.

❆✜r♠❛♠♦s q✉❡ ❡st❛ r❡♣r❡s❡♥t❛çã♦ é tr✐✈✐❛❧ ❡♠r(g)✳ ❉❡ ❢❛t♦✱ s❡ v ∈V✱ X ∈r(g) ❡w∈L✱ t❡♠♦s

(i)✿ Xv⊗ψ(w) =eλ(X)v⊗ψ(w) ❡

(ii)✿ (Xψ)(w) =−ψ(Xw) = −ψ(eλ(X)w) = −eλ(X)ψ(w) ⇒v⊗(Xψ)(w) =

−eλ(X)v⊗ψ(w).

❉❡ (i) ❡ (ii)✱ s❡❣✉❡ ❛ ❛✜r♠❛çã♦✳ P♦rt❛♥t♦✱ g×(V0⊗L) → (V0 ⊗L)✱

♦♥❞❡ V0 =V ⊗L∗✱ é ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ g♣r♦❝✉r❛❞❛✳

❆ ♣❛rt✐r ❞❡st❛ ♣r♦♣♦s✐çã♦✱ ♣❡r❝❡❜❡♠♦s ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦ ❡st✉❞♦ ❞❛s r❡♣r❡s❡♥t❛çõ❡s ✐rr❡❞✉tí✈❡✐s ❞❛s á❧❣❡❜r❛s ❞❡ ▲✐❡ s❡♠✐ss✐♠♣❧❡s✱ ♣♦✐s t♦❞❛ r❡♣r❡✲ s❡♥tçã♦ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ q✉❛❧q✉❡r s❡ ❞❡❝♦♠♣õ❡ ❡♠ ✉♠❛ ♣❛rt❡ s♦❧ú✈❡❧ ✭L✮ ❡ ♦✉tr❛ ✐rr❡❞✉tí✈❡❧ ✭V0✮✳ ◆♦t❡ ❛✐♥❞❛ q✉❡ ♦ ❚❡♦r❡♠❛ ❞❡ ▲✐❡ ♥♦s ❞✐③ ✉♠❛

✐♥❢♦r♠❛çã♦ ✐♠♣♦rt❛♥t❡ ❛ r❡s♣❡✐t♦ ❞❛s r❡♣r❡s❡♥t❛çõ❡s ❞❡ g ⊂ gl(V) q✉❛♥❞♦ g é s♦❧ú✈❡❧ ❡ V t❡♠ ❞✐♠❡♥sã♦ ✜♥✐t❛✿ ♥❡st❡ ❝❛s♦✱ V ♣♦ss✉✐ ✉♠ s✉❜❡s♣❛ç♦ g✲✐♥✈❛r✐❛♥t❡ W =hvi✱ ♦✉ ❞❡ ❢♦r♠❛ ❡q✉✐✈❛❧❡♥t❡✱ t♦❞❛ r❡♣r❡s❡♥t❛çã♦ ✐rr❡❞✉tí✲

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✶✸

✷ ➪❧❣❡❜r❛s ❞❡ ▲✐❡ ❙❡♠✐ss✐♠♣❧❡s

◆❡st❛ s❡çã♦✱ ❝♦♥s✐❞❡r❡ ♦ ❝♦r♣♦ ❜❛s❡ K ❝♦♠ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦✱ ❛s

á❧❣❡❜r❛s ❞❡ ▲✐❡ ❡ ♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛✳ ❱✐♠♦s q✉❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ g é s❡♠✐s✐♠♣❧❡s s❡ s❡✉ r❛❞✐❝❛❧ r(g) é ③❡r♦✳ ❆❣♦r❛✱ ♥♦ss♦

♦❜❥❡t✐✈♦ é ❛♣r❡s❡♥t❛r ❝r✐tér✐♦s ♣❛r❛ s❛❜❡r♠♦s s❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ é s❡♠✐ss✐♠♣❧❡s✳

✷✳✶ ❈r✐tér✐♦ ❞❡ s❡♠✐ss✐♠♣❧✐❝✐❞❛❞❡

❉❡✜♥✐çã♦ ✷✳✶✳✶✳ ❙❡❥❛ g ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✳ ❆ ❢♦r♠❛ ❜✐❧✐♥❡❛r s✐♠étr✐❝❛ B :g×g→K ❞❛❞❛ ♣♦r B(X, Y) =tr(❛❞(X)◦❛❞(Y))é ❝❤❛♠❛❞❛ ❢♦r♠❛ ❞❡

❑✐❧❧✐♥❣✳

❊①❡♠♣❧♦ ✷✳✶✳✶✳ ❙❛❜❡♠♦s q✉❡ s❡{e1, . . . , en}é ✉♠❛ ❜❛s❡ ♣❛r❛g✱ ❡♥tã♦ t♦❞❛

❢♦r♠❛ ❜✐❧✐♥❡❛r ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦ B(X, Y) = Xt_BY_{✱ ♦♥❞❡} _B _{é ❛ ♠❛tr✐③} B(ei, ej)ij ❡ Xt é ❛ tr❛♥s♣♦st❛ ❞❡ X✳ ❈♦♥s✐❞❡r❡ ❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ sl(2,C)✳

➱ ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡ s❡ t♦♠❛r♠♦s ❛ ❜❛s❡ ♦r❞❡♥❛❞❛ {X, H, Y} ❞❡ sl(2,C)

♠❡♥❝✐♦♥❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡✱ ♦❜t❡♠♦s

❛❞(X) =

 

0 −2 0 0 0 1 0 0 0



_{✱ ❛❞}(H) =

 

2 0 0 0 0 0 0 0 −2



_{❡ ❛❞}(Y) =

 

0 0 0

−1 0 0 0 2 0

 _✳

▲♦❣♦✱ s❡ e1 =X, e2 =H ❡ e3 =Y✱ ❡♥tã♦ B =





0 0 4 0 8 0 4 0 0



✳

Pr♦♣♦s✐çã♦ ✷✳✶✳✶✳ ❆ ❢♦r♠❛ ❞❡ ❑✐❧❧✐♥❣ B ♣♦ss✉✐ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

(a) B é ❛ss♦❝✐❛t✐✈❛✿ B([X, Y], Z) =B(X,[Y, Z])✳

(b) B é ✐♥✈❛r✐❛♥t❡✿ B(D(X), Y) +B(X, D(Y)) = 0 ♣❛r❛ t♦❞❛ ❞❡r✐✈❛çã♦D

❡♠ g✳

(c) ❙❡ θ ∈Aut(g)✱ ❡♥tã♦ B(θ(X), θ(Y)) =B(X, Y) ♣❛r❛ t♦❞♦ X, Y ∈g✳

❉❡♠♦♥str❛çã♦✳ P❛r❛(a)✱ ❜❛st❛ ✉s❛r q✉❡ ♦ tr❛ç♦ é ❝í❝❧✐❝♦✱ ♦✉ s❡❥❛✱ tr(XY Z) =

tr(ZXY) = tr(Y ZX) ♣❛r❛ q✉❛✐sq✉❡r ❡♥❞♦♠♦r✜s♠♦s X, Y, Z✳ ❊♠ (b)✱

♦❜s❡r✈❡ q✉❡ ♣❛r❛ t♦❞♦ X ∈g t❡♠✲s❡ ❛❞(D(X)) =D ◦❛❞(X)− ❛❞(X) ◦D✳

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✷✳✶ ❈r✐tér✐♦ ❞❡ s❡♠✐ss✐♠♣❧✐❝✐❞❛❞❡ ✶✹

Pr♦♣♦s✐çã♦ ✷✳✶✳✷✳ ❈♦♥s✐❞❡r❡ B ❛ ❢♦r♠❛ ❞❡ ❑✐❧❧✐♥❣ ❞❡ g✳ ❙❡ a é ✉♠ ✐❞❡❛❧

❞❡ g✱ ❡♥tã♦ Ba=B|a×a✱ ♦♥❞❡ Ba é ❛ ❢♦r♠❛ ❞❡ ❑✐❧❧✐♥❣ ❞❡ a✳

❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ♣r❡❝✐s❛r ❞♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❞❛ ➪❧❣❡❜r❛ ▲✐♥❡❛r✿ ❙❡

W ⊂ V é ✉♠ s✉❜❡s♣❛ç♦✱ ♦♥❞❡ V t❡♠ ❞✐♠❡♥sã♦ ✜♥✐t❛ ❡ f : V → V ✉♠

❡♥❞♦♠♦r✜s♠♦ t❛❧ q✉❡ f(V) ⊂ W✱ ❡♥tã♦ tr(f) = tr(f|W) ✭♣❛r❛ ✈❡r ✐st♦✱

❡st❡♥❞❛ ✉♠❛ ❜❛s❡ ❞❡ W ❛ ✉♠❛ ❜❛s❡ ❞❡ V ❡ ❝♦♥s✐❞❡r❡ ❛ ♠❛tr✐③ ❞❡ f ♥❡st❛

❜❛s❡✮✳ ❙❡♥❞♦ ❛ss✐♠✱ t♦♠❡ X, Y ∈ a✳ ▲♦❣♦✱ ❛❞(X) ◦ ❛❞(Y) : g → g é ✉♠

❡♥❞♦♠♦r✜s♠♦ t❛❧ q✉❡ ✭❛❞(X)◦ ❛❞(Y))(g)⊂a ❡ ♣❡❧♦ r❡s✉❧t❛❞♦ ♠❡♥❝✐♦♥❛❞♦

❛❝✐♠❛✱

tr✭❛❞(X)◦ ❛❞(Y)) = tr((❛❞(X) ◦ ❛❞(Y))|a) = tr(❛❞(X)|a ◦ ❛❞(Y)|a)✳

❉❡✜♥✐çã♦ ✷✳✶✳✷✳ ❉❡✜♥✐♠♦s ♦ r❛❞✐❝❛❧ ❞❡ ✉♠❛ ❢♦r♠❛ ❜✐❧✐♥❡❛r s✐♠étr✐❝❛ ❛r❜✐trár✐❛ f ❡♠ g ♣♦r ❘❛❞(f) ={X ∈ g | f(X, Y) = 0, ∀Y ∈g} ❡ ❞✐③❡♠♦s

q✉❡ ❛ ❢♦r♠❛ f é ♥ã♦✲❞❡❣❡♥❡r❛❞❛ s❡ ❘❛❞(f) = 0✱ ♦✉ ❞❡ ❢♦r♠❛ ❡q✉✐✈❛❧❡♥t❡✱

❞❡t f 6= 0✳

◆♦t❡ q✉❡ ❘❛❞(B) é ✐❞❡❛❧ ❞❡ g✱ ♣♦✐s s❡ X ∈ ❘❛❞(B) ❡ Y ∈ g✱ ❡♥✲

tã♦ B([X, Y], Z) = B(X,[Y, Z]) = 0 ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ B é ❛ss♦❝✐❛t✐✈❛ ❡ X ∈ ❘❛❞(B)✳

❖❜s❡r✈❛çã♦ ✷✳✶✳✶✳ ❯♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ g é s❡♠✐ss✐♠♣❧❡s s❡✱ ❡ só s❡✱ g só

♣♦ss✉✐ ♦ ✐❞❡❛❧ ❛❜❡❧✐❛♥♦ tr✐✈✐❛❧ 0✳ ❚♦♠❡g s❡♠✐ss✐♠♣❧❡s ❡a ✉♠ ✐❞❡❛❧ ❛❜❡❧✐❛♥♦

❞❡g✭❡♠ ♣❛rt✐❝✉❧❛r✱aé s♦❧ú✈❡❧✮✱ ❧♦❣♦a⊆r(g) = 0✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛

q✉❡ g ♥ã♦ é s❡♠✐ss✐♠♣❧❡s ✭r=r(g)6= 0✮✳ ❈♦♠♦ r é s♦❧ú✈❡❧✱ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ n t❛❧ q✉❡ r(n) = 0❡ r(n−1) 6= 0✳ ▲♦❣♦✱ r(n−1) é ✉♠ ✐❞❡❛❧ ❛❜❡❧✐❛♥♦ ♥ã♦✲♥✉❧♦ ❞❡ g✳

❚❡♦r❡♠❛ ✷✳✶✳✶ ✭❈r✐tér✐♦ ❞❡ ❈❛rt❛♥✮✳ ❙❡❥❛ g s✉❜á❧❣❡❜r❛ ❞❡ gl(V)✱ ♦♥❞❡ V

t❡♠ ❞✐♠❡♥sã♦ ✜♥✐t❛✳ ❊♥tã♦ g é s♦❧ú✈❡❧ s❡✱ ❡ só s❡✱ tr(XY) = 0 ♣❛r❛ t♦❞♦ X ∈[g,g] ❡ Y ∈g✳

❉❡♠♦♥str❛çã♦✳ ✭⇒✮ ❙❡gé s♦❧ú✈❡❧✱ ❡①✐st❡ ✉♠❛ ❜❛s❡ ♣❛r❛V t❛❧ q✉❡ ♣❛r❛ t♦❞♦

❡❧❡♠❡♥t♦ ❞❡ g é r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠❛ ♠❛tr✐③ tr✐❛♥❣✉❧❛r s✉♣❡r✐♦r✳ ▲♦❣♦✱ s❡ X ∈[g,g]✱ ❡♥tã♦Xé ✉♠❛ ♠❛tr✐③ ❡str✐t❛♠❡♥t❡ tr✐❛♥❣✉❧❛r ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ XY t❛♠❜é♠ ♦ é✳

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✷✳✷ ❚❡♦r❡♠❛ ❞❡ ❲❡②❧✱ ❡❧❡♠❡♥t♦s s❡♠✐ss✐♠♣❧❡s ❡ ♥✐❧♣♦t❡♥t❡s ✶✺

❈♦r♦❧ár✐♦ ✷✳✶✳✶✳ ❙❡❥❛ g ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ t❛❧ q✉❡ tr(❛❞(X) ◦ ❛❞(Y)) = 0

♣❛r❛ t♦❞♦ X ∈[g,g] ❡ Y ∈g✳ ❊♥tã♦✱ g é s♦❧ú✈❡❧✳

❉❡♠♦♥str❛çã♦✳ P♦r ❤✐♣ót❡s❡✱ tr(❛❞(X) ◦ ❛❞(Y)) = 0 ♣❛r❛ t♦❞♦ X ∈ [g,g]

❡ Y ∈ g✱ ♦✉ s❡❥❛✱ ♣❛r❛ t♦❞♦ ❛❞(X) ∈ [❛❞(g),❛❞(g)] ❡ ❛❞(Y) ∈ ❛❞(g)✳ P❡❧♦

❈r✐tér✐♦ ❞❡ ❈❛rt❛♥✱ ❛❞(g) é s♦❧ú✈❡❧✳ P♦r ♦✉tr♦ ❧❛❞♦✱ g/z(g) ∼= ❛❞(g) ✭❚❡♦✲

r❡♠❛ ❞♦ ■s♦♠♦r✜s♠♦✮ s❡♥❞♦ ❛❞(g) ❡ z(g) s♦❧ú✈❡✐s✳ P♦rt❛♥t♦ g é s♦❧ú✈❡❧ ♣❡❧❛

Pr♦♣♦s✐çã♦ ✶✳✸✳✶✳

❚❡♦r❡♠❛ ✷✳✶✳✷✳ ❯♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ é s❡♠✐ss✐♠♣❧❡s s❡✱ ❡ só s❡✱ s✉❛ ❢♦r♠❛ ❞❡ ❑✐❧❧✐♥❣ é ♥ã♦✲❞❡❣❡♥❡r❛❞❛✳

❉❡♠♦♥str❛çã♦✳ ✭⇒✮ ❙✉♣♦♥❤❛ B ❞❡❣❡♥❡r❛❞❛✱ ♦✉ s❡❥❛✱ ❡①✐st❡ X ∈g ♥ã♦✲♥✉❧♦

t❛❧ q✉❡X ∈R=❘❛❞(B)✳ ❉❛í✱R é ✉♠ ✐❞❡❛❧ ♥ã♦✲♥✉❧♦ ❞❡g✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ R ❡ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✶✳✷✱ BR =B|R×R = 0✳ ❆ss✐♠✱ tr(❛❞(X)◦❛❞(Y)) = 0

♣❛r❛ t♦❞♦ X, Y ∈ R✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ t♦❞♦ X ∈ [R, R] ⊂ R ❡ Y ∈ R✳

P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♥s✐❞❡r❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❛❞❥✉♥t❛ ❛❞|R : R → ❛❞(R) ⊂ gl(R) ❡ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ ■s♦♠♦r✜s♠♦✱ R/z(R) ∼= ❛❞(R)✳ ❆♦ ♠❡s♠♦ t❡♠♣♦✱

❛❞(R)é s♦❧ú✈❡❧ ✭❈r✐tér✐♦ ❞❡ ❈❛rt❛♥✮✳ ❆ss✐♠✱ ♣❡❧♦ ✐s♦♠♦r✜s♠♦ ❛♥t❡r✐♦r ❡ ♣❡❧❛

Pr♦♣♦s✐çã♦ ✶✳✸✳✶✱ Ré s♦❧ú✈❡❧ ✭z(R)é s♦❧ú✈❡❧✮ ❝♦♠R ⊂r(g)✳ P♦rt❛♥t♦✱ g♥ã♦

é s❡♠✐ss✐♠♣❧❡s✳

✭⇐✮ ❙❡ g ♥ã♦ é s❡♠✐ss✐♠♣❧❡s✱ g ♣♦ss✉✐ ✉♠ ✐❞❡❛❧ ❛❜❡❧✐❛♥♦ ♥ã♦✲♥✉❧♦ a✳ ❚♦♠❡ X ∈a ♥ã♦✲♥✉❧♦ ❡ Y ∈g✱ ❧♦❣♦ (❛❞(X)◦ ❛❞(Y))2 _{= 0}_{✱ ✐st♦ é✱ ❛❞}_(X) _◦ _❛❞_(Y₎

é ✉♠ ♦♣❡r❛❞♦r ♥✐❧♣♦t❡♥t❡ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♣♦ss✉✐ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♠❛tr✐❝✐❛❧ ❞❡ tr❛ç♦ ③❡r♦✳ ■st♦ ✐♠♣❧✐❝❛ q✉❡ B(X, Y) = 0 ♣❛r❛ t♦❞♦ X ∈ a ❡ Y ∈g ✭♦ tr❛ç♦ ✐♥❞❡♣❡♥❞❡ ❞❛ ❜❛s❡✮✳ P♦rt❛♥t♦✱ a⊂R ❡B é ❞❡❣❡♥❡r❛❞❛✳

P❡❧♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✱ ❘❛❞(B) ⊂ r(g) ❡ ✜❝❛ ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡ sl(2,C)

é s❡♠✐ss✐♠♣❧❡s✱ ♣♦✐s ❞❡t B =−128✳

❚❡♦r❡♠❛ ✷✳✶✳✸✳ ❙❡ g é s❡♠✐ss✐♠♣❧❡s✱ ❡♥tã♦ t♦❞❛s ❛s ❞❡r✐✈❛çõ❡s ❞❡ g sã♦

✐♥t❡r♥❛s✳

❉❡♠♦♥str❛çã♦✳ ❱❡❥❛ ❬✷❪ ✭♣á❣✐♥❛ ✷✸✮✳

✷✳✷ ❚❡♦r❡♠❛ ❞❡ ❲❡②❧✱ ❡❧❡♠❡♥t♦s s❡♠✐ss✐♠♣❧❡s ❡ ♥✐❧♣♦✲

t❡♥t❡s

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✷✳✷ ❚❡♦r❡♠❛ ❞❡ ❲❡②❧✱ ❡❧❡♠❡♥t♦s s❡♠✐ss✐♠♣❧❡s ❡ ♥✐❧♣♦t❡♥t❡s ✶✻

❚❡♦r❡♠❛ ✷✳✷✳✶ ✭❲❡②❧✮✳ ❚♦❞❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ s❡♠✐s✲ s✐♠♣❧❡s é ❝♦♠♣❧❡t❛♠❡♥t❡ r❡❞✉tí✈❡❧✳

❉❡♠♦♥str❛çã♦✳ ❱❡❥❛ ❬✷❪ ✭♣á❣✐♥❛ ✷✽✮✳

❘❡❧❡♠❜r❡ q✉❡ ♦ ❚❡♦r❡♠❛ ❞❛ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❏♦r❞❛♥ ♥♦s ❞✐③ q✉❡ t♦❞♦ ❡♥❞♦♠♦r✜s♠♦ X ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❞❡ ❢♦r♠❛

ú♥✐❝❛ ❝♦♠♦ X = Xs +Xn✱ ♦♥❞❡ Xs é ❞✐❛❣♦♥❛❧✐③á✈❡❧✱ Xn é ♥✐❧♣♦t❡♥t❡ ❡ Xs, Xn ❝♦♠✉t❛♠✳ ❆ss✐♠✱ ♣❛r❛ g∈gl(V) ✭V ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦✮ t♦❞♦

❡❧❡♠❡♥t♦ X ∈ g ♣♦❞❡ s❡r ❡s❝r✐t♦ ❞❡ ❢♦r♠❛ ú♥✐❝❛ X = Xs+Xn✳ ❊♠ ❣❡r❛❧✱ Xs ❡ Xn sã♦ ❡❧❡♠❡♥t♦s ❡♠ gl(V)✱ ♠❛s ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❡♠ g✳

Pr♦♣♦s✐çã♦ ✷✳✷✳✶✳ ❙❡ g⊂gl(V) é s❡♠✐ss✐♠♣❧❡s✱ ❡♥tã♦ Xs, Xn∈g✳

❉❡♠♦♥str❛çã♦✳ ❱❡❥❛ ❬✶❪ ✭♣á❣✐♥❛ ✹✽✷✮✳

❆❣♦r❛✱ s❡ ❝♦♥s✐❞❡r❛r♠♦sg✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✱X ∈g❡ρ:g→gl(C, n)

✉♠❛ r❡♣r❡s❡♥t❛çã♦✱ q✉❡r❡♠♦s s❛❜❡r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ρ(X) ❝♦♠ r❡s♣❡✐t♦

❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ❏♦r❞❛♥✳ P♦r ❡①❡♠♣❧♦✱ s❡ g=C✱ ❝♦♥s✐❞❡r❡ ♦s ❝❛s♦s (i) ♥❛ r❡♣r❡s❡♥t❛çã♦ ρ1 : t 7→ (t)1×1 t♦❞♦ ❡❧❡♠❡♥t♦ é ❞✐❛❣♦♥❛❧✐③á✈❡❧✱ ✐st♦ é✱

ρ1(X)s =ρ1(X).

(ii) s❡ρ2 :t 7→

0 t 0 0

✱ ❡♥tã♦ t♦❞♦ ❡❧❡♠❡♥t♦ é ♥✐❧♣♦t❡♥t❡ ✭ρ2(X)s= 0✮✳

(iii) ❡ s❡ ρ3 : t 7→

t t 0 0

=

t 0 0 0

+

0 t 0 0

✱ ρ3(X) ♥ã♦ é ❞✐❛❣♦♥❛❧✐③á✈❡❧

♥❡♠ ♥✐❧♣♦t❡♥t❡❀ ❛ ♣❛rt❡ ❞✐❛❣♦♥❛❧✐③á✈❡❧ ❡ ♥✐❧♣♦t❡♥t❡ ❞❡ ρ3(X) ♥ã♦ ♣❡rt❡♥❝❡♠

❛ ✐♠❛❣❡♠ ρ(g)✳

▼❛s✱ q✉❛♥❞♦ g é s❡♠✐ss✐♠♣❧❡s✱ ❛ s✐t✉❛çã♦ é ❜❛st❛♥t❡ ❞✐❢❡r❡♥t❡ ❝♦♠♦

♠♦str❛ ♦ ♣ró①✐♠♦ t❡♦r❡♠❛✿

❚❡♦r❡♠❛ ✷✳✷✳✷ ✭Pr❡s❡r✈❛çã♦ ❞❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ❏♦r❞❛♥✮✳ ❙❡❥❛g✉♠❛ á❧❣❡✲

❜r❛ ❞❡ ▲✐❡ s❡♠✐ss✐♠♣❧❡s✳ P❛r❛ t♦❞♦ X ∈g✱ ❡①✐st❡♠ Xs, Xn ∈g t❛✐s q✉❡ ♣❛r❛

t♦❞❛ r❡♣r❡s❡♥t❛çã♦ ρ:g→gl(V) t❡♠✲s❡ ρ(X)s =ρ(Xs) ❡ ρ(X)n=ρ(Xn)✳

❉❡♠♦♥str❛çã♦✳ ❱❡❥❛ ❬✶❪ ✭♣á❣✐♥❛ ✹✽✸✮✳

❈♦♥s✐❞❡r❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❛❞❥✉♥t❛ ❛❞: g → gl(g) ❞❡ ✉♠❛ á❧❣❡❜r❛ ❞❡

▲✐❡ s❡♠✐ss✐♠♣❧❡s✳ ❈♦♠♦ ❑❡r✭❛❞✮=z(g)é ✉♠ ✐❞❡❛❧ s♦❧ú✈❡❧ ❞❡ g✱ ♦❜t❡♠♦s q✉❡

❑❡r✭❛❞✮=z(g)⊂r(g) = 0✱ ♦✉ s❡❥❛✱ ❛❞ é ✐♥❥❡t✐✈❛ ✭❡st❡ é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦