Biểu diễn bất khả quy và phân loại đại số lie nửa đơn

❇é ●✐➳♦ ❉ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❱✐♥❤ ❚r➬♥ ❍➯✐ ◆❤➞♥ ❇✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ✈➭ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ ▲✉❐♥ ✈➝♥ t❤➵❝ sÜ t♦➳♥ ❤ä❝ ◆❣❤Ư ❆♥ ✲ ✷✵✶✼ ❇é ●✐➳♦ ❉ơ❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❱✐♥❤ ❚r➬♥ ❍➯✐ ◆❤➞♥ ❇✐Ó✉ ❞✐Ô♥ ❜✃t ❦❤➯ q✉② ✈➭ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ ▲✉❐♥ ✈➝♥ t❤➵❝ sÜ t♦➳♥ ❤ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ➜➵✐ sè ✈➭ ▲ý t❤✉②Õt sè ▼➲ sè✿ ✻✷ ✹✻ ✵✶ ✵✹ ◆❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ ❚❙✳ ◆❣✉②Ơ♥ ◗✉è❝ ❚❤➡ ◆❣❤Ư ❆♥ ✲ ✷✵✶✼ ▼ơ❝ ❧ơ❝ ▲ê✐ ♥ã✐ ➤➬✉ ✸ ✶ ✼ ✷ ➜➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ ✶✳✶ ➜➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❇✐Ĩ✉ ❞✐Ơ♥ ❝đ❛ s❧2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✸ ➜➵✐ sè ❝♦♥ ❈❛rt❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✹ ❉➵♥❣ ❝♦♠♣❛❝t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✺ ▼ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❤Ư ♥❣❤✐Ư♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ❇✐Ó✉ ❞✐Ô♥ ❜✃t ❦❤➯ q✉② ✈➭ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ❝ã ❝❤✐Ị✉ t❤✃♣ ✸✷ ✷✳✶ ❚rä♥❣ ✈➭ ✈❡❝t➡ trä♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✷ ❙ù ❤♦➭♥ t♦➭♥ ❦❤➯ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✸ P❤➞♥ ❧♦➵✐ ❝➳❝ ➤➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✹ ❇✐Ó✉ ❞✐Ô♥ trù❝ ❣✐❛♦ ✈➭ ➤è✐ ♥❣➱✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✺ ❇✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝ñ❛ ✹✻ sl2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ❑Õt ❧✉❐♥ ✹✾ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✺✵ ✷ ▲ê✐ ♥ã✐ ➤➬✉ ✶✳ ▲ý ❞♦ ❝❤ä♥ ➤Ò t➭✐ ◆❤➢ t❛ ➤➲ ❜✐Õt ❙♦♣❤✉s ▲✐❡ ✭✶✼✴✶✷✴✶✽✹✷ ✲ ✶✽✴✵✷✴✶✽✾✾✮ ❧➭ ♥❣➢ê✐ t➵♦ r❛ ❧ý t❤✉②Õt ❝đ❛ ❝➳❝ ➤è✐ ①ø♥❣ ❧✐➟♥ tơ❝ ✈➭ ✈❐♥ ❞ơ♥❣ ♥ã ✈➭♦ ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ ❤×♥❤ ❤ä❝✱ ♣❤➢➡♥❣ tr×♥❤ ➤➵♦ ❤➭♠ r✐➟♥❣ ✈➭ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥✳ ❈➠♥❣ ❝ơ ❝❤Ý♥❤ ❝đ❛ ▲✐❡ ✈➭ ♠ét tr♦♥❣ ♥❤÷♥❣ t❤➭♥❤ tù✉ ✈Ü ➤➵✐ ♥❤✃t ❝ñ❛ ➠♥❣ ❧➭ sù ❦❤➳♠ ♣❤➳ r❛ ❝➳❝ ♥❤ã♠ ❜✐Õ♥ ➤ỉ✐ ❧✐➟♥ tơ❝✱ ♠➭ ♥❣➭② ♥❛② ♥❣➢ê✐ t❛ ❣ä✐ t❤❡♦ t➟♥ ➠♥❣ ❧➭ ♥❤ã♠ ▲✐❡✳ ◆❤ã♠ ▲✐❡ ❧➭ ❦❤➳✐ ♥✐Ư♠ tỉ♥❣ ❤ß❛ tõ ❤❛✐ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ❧➭ ♥❤ã♠ ✭tr♦♥❣ ➜➵✐ sè✮ ✈➭ ➤❛ t➵♣ ✈✐ ♣❤➞♥ ✭tr♦♥❣ ì ọ ó ợ ứ ụ ❧ý t❤✉②Õt ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥✱ ♥ã ➤➲ ❝✉♥❣ ❝✃♣ ♠ét ♣❤➢➡♥❣ t✐Ư♥ tù ♥❤✐➟♥ ➤Ĩ ♣❤➞♥ tÝ❝❤ ❝➳❝ ➤è✐ ①ø♥❣ ❧✐➟♥ tơ❝ ❝đ❛ ❝➳❝ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥✱ tr♦♥❣ ột tứ ó ị ợ sử ❞ơ♥❣ tr♦♥❣ ❧ý t❤✉②Õt ●❛❧♦✐s ➤Ĩ ♣❤➞♥ tÝ❝❤ ❝➳❝ ➤è✐ ứ rờ r ủ trì số ỗ ❜✐Ĩ✉ t❤ø❝ ❜✃t ❜✐Õ♥ ❞➢í✐ t➳❝ ➤é♥❣ ❝đ❛ ♥❤ã♠ ▲✐❡ ❝❤♦ ♠ét tÝ❝❤ ♣❤➞♥ ➤➬✉ ❝đ❛ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥✱ ❞♦ ✈❐② ❝❤♦ ♣❤Ð♣ t❛ ❤➵ ❜❐❝ ❝đ❛ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ➤ã✳ ◆❤ã♠ ▲✐❡ ❦❤➠♥❣ ❝❤Ø ❧➭ ❝➠♥❣ ❝ơ ❝đ❛ ❣➬♥ ♥❤➢ t✃t ❝➯ ❝➳❝ ♥❣➭♥❤ ❚♦➳♥ ❤✐Ư♥ ➤➵✐✱ ♠➭ ♥ã ❝ß♥ ❧➭ ❝➠♥❣ ❝ơ ➤Ĩ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ♥❣➭♥❤ ❝đ❛ ❱❐t ❧ý ❧ý t❤✉②Õt ❤✐Ư♥ ➤➵✐✳ ▼ét tr♦♥❣ ♥❤÷♥❣ ý t➢ë♥❣ ❝đ❛ ❧ý t❤✉②Õt ♥❤ã♠ ▲✐❡ ❧➭ t❤❛② t❤Õ ❝✃✉ tró❝ ♥❤ã♠ t♦➭♥ ❝ơ❝ ❜ë✐ ♣❤✐➟♥ tí ị ủ ó ò ọ ❧➭ ♣❤✐➟♥ ❜➯♥ ➤➲ ➤➢ỵ❝ ❧➭♠ t✉②Õ♥ tÝ♥❤ ❤ã❛✳ ❙✳ ▲✐❡ ❣ä✐ ➤ã ❧➭ ♥❤ã♠ ▲✐❡ ✈➠ ❝ï♥❣ ❜Ð✳ ❙❛✉ ♥➭② ♥❣➢ê✐ t❛ ❣ä✐ ➤ã ❧➭ ➜➵✐ sè ▲✐❡✳ ◆❣❤✐➟♥ ❝ø✉ ✈Ị ➤➵✐ sè ▲✐❡ ✈➭ ❧ý t❤✉②Õt ❜✐Ĩ✉ ❞✐Ơ♥ ❝đ❛ ➤➵✐ sè ▲✐❡ ♥ã✐ ❝❤✉♥❣✱ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ✈➭ ❜✐Ĩ✉ ❞✐Ơ♥ t➢➡♥❣ ø♥❣ ❝đ❛ ♥ã ♥ã✐ r✐➟♥❣ ❧➭ ♠ét ❧Ü♥❤ ✈ù❝ ♥❣❤✐➟♥ ❝ø✉ ré♥❣ tr♦♥❣ ❚♦➳♥ ❤ä❝ ✈➭ ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ tr♦♥❣ ❝➳❝ ♥❣➭♥❤ ❦❤♦❛ ❤ä❝ ột tr ữ t ợ ề q✉❛♥ t➞♠ ✈Ị ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ❧➭ ①➞② ❞ù♥❣ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉②✱ ♣❤➞♥ ❧♦➵✐ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ✈➭ ❝➳❝ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥✳ ✸ ✹ ▲ê✐ ♥ã✐ ➤➬✉ ◆➝♠ ✷✵✶✵✱ ❞♦ ♥❤✉ ❝➬✉ ❜✐Ó✉ ❞✐Ơ♥ ♥❣❤✐Ư♠ ❝đ❛ ♠ét ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤ t❤✉➬♥ ♥❤✃t t❤➠♥❣ q✉❛ ♥❣❤✐Ư♠ ❝đ❛ ❝➳❝ ♣❤➢➡♥❣ tr×♥❤ t✉②Õ♥ tÝ♥❤ ❜❐❝ t❤✃♣ ❤➡♥ ✈➭ t❤➠♥❣ q✉❛ ❧ý t❤✉②Õt ●❛❧♦✐s ✈✐ ♣❤➞♥✱ ❤❛✐ t➳❝ ❣✐➯ ◆❣✉②Ô♥ ❆♥ ❑❤➢➡♥❣ ✈➭ ▼❛r✐✉s ✈❛♥ ❞❡ P✉t ✭①❡♠ ❬✺❪✮ ➤➲ ❞ï♥❣ ♣❤➬♥ ♠Ò♠ trù❝ t✉②Õ♥ ▲✐❡ ✭①❡♠ ❬✸❪✮ ➤Ó tÝ♥❤ t♦➳♥ ✈➭ t❤✐Õt ❧❐♣ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝ã sè ❝❤✐Ị✉ ❦❤➠♥❣ q✉➳ 11 ❝❤♦ ❝➳❝ ➤➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ ❝ã ❝❤✐Ị✉ t❤✃♣✳ ❇➯♥❣ ❦Õt q✉➯ ♥➭② ❝➭♥❣ ❧í♥ t❤× t❛ ó tể rộ ợ trì ✈✐ ♣❤➞♥ t✉②Õ♥ tÝ♥❤ t❤✉➬♥ ♥❤✃t ❝ã ♥❣❤✐Ư♠ ❝ã t❤Ĩ ể ễ ợ t q ệ ủ trì ❝ã ❜❐❝ t❤✃♣ ❤➡♥✳ ❱✐Ư❝ ♠➠ t➯ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝đ❛ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ✈➭ ❜➭✐ t♦➳♥ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ ➤❛♥❣ ➤➢ỵ❝ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ tr♦♥❣ ✈➭ ♥❣♦➭✐ ♥➢í❝ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉✳ ❱í✐ ♠♦♥❣ ♠✉è♥ t×♠ ❤✐Ĩ✉ ✈Ị ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ✈➭ ❜✐Ĩ✉ ❞✐Ơ♥ ❝đ❛ ♥ã✱ ❝❤ó♥❣ t➠✐ ❝❤ä♥ ➤Ị t➭✐✿ ❇✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ✈➭ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ❧➭♠ ➤Ị t➭✐ ❧✉❐♥ ✈➝♥✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ❧➭ ❞ù❛ tr➟♥ ❜➭✐ ❜➳♦ ✧❙❧♦✈✐♥❣ ▲✐♥❡❛r ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✧✱ P✉r❡ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ◗✉❛rt❡r❧②✱ ❱♦❧✉♠❡ ✻✱ ◆✉♠❜❡r ✶✭✷✵✶✵✮✱ ♣♣✳ ✶✼✸ ✲ ✷✵✽ ❝ñ❛ ❝➳❝ t➳❝ ❣✐➯ ◆❣✉②Ơ♥ ❆♥ ❑❤➢➡♥❣✱ ▼❛r✐✉s ✈❛♥ ❞❡ P✉t ➤Ĩ tì ể tí tổ ợ trì ♠ét ❝➳❝❤ ❤Ư t❤è♥❣ ❝➳❝ ❦Õt q✉➯ ✈Ị ❜✐Ĩ✉ ❞✐Ơ♥ ✈➭ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥✳ ❈✉è✐ ❝ï♥❣ ❞ï♥❣ ♣❤➬♥ ♠Ò♠ trù❝ t✉②Õ♥ ▲✐❡ ➤ã ❧➭ ▲✐❊ ❖♥❧✐♥❡ ❙❡r✈✐❝❡ ❝đ❛ ❆✳ ▼✳ ❈♦❤❡♥ tr➟♥ ❲❡❜s✐t❡ ❤tt♣✿✴✴✇✇✇✲♠❛t❤✳✉♥✐✈✲♣♦✐t✐❡rs✳❢r✴ ♠❛❛✈❧✴▲✐❊✴ ➤Ĩ tÝ♥❤ ❧➵✐ ❜➯♥❣ ❦Õt q✉➯ tr➟♥ ❝❤♦ ❝➳❝ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ❝ã sè ❝❤✐Ị✉ t❤✃♣✳ ✷✳ ◆é✐ ❞✉♥❣ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ ❧✉❐♥ ✈➝♥ ✷✳✶✳ ❚r×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝ñ❛ ➤➵✐ sè ▲✐❡ ✈➭ ❧ý t❤✉②Õt ❝✃✉ tró❝ ❝đ❛ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥✱✳✳✳ ✷✳✷✳ ❚r×♥❤ ❜➭② ❦Õt q✉➯ ✈Ị ❜✐Ĩ✉ ❞✐Ơ♥ ❝đ❛ sl2 , ♠➠ t➯ ❜✐Ĩ✉ ❞✐Ơ♥ ❝đ❛ ♥ã t❤❡♦ trä♥❣ ✈➭ ❤Ư ♥❣❤✐Ư♠✳ ❚✐Õ♣ t❤❡♦ tr×♥❤ ❜➭② ❧ý t❤✉②Õt ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝đ❛ ❝➳❝ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ t❤❡♦ trä♥❣ ✈➭ ✈❡❝t➡ trä♥❣✱ tõ ➤ã ♣❤➞♥ ❧♦➵✐ ❝➳❝ ➤➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥✳ ✷✳✸✳ ❈✉è✐ ❝ï♥❣ ❞ï♥❣ ♣❤➬♥ ♠Ị♠ trù❝ t✉②Õ♥ ▲✐❡ ➤Ĩ tÝ♥❤ t♦➳♥ ✈➭ ♣❤➞♥ ❧♦➵✐ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝đ❛ ❝ã ❝❤✐Ị✉ ❦❤➠♥❣ q✉➳ 11 ❝❤♦ ➤➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ ❝ã sè ❝❤✐Ị✉ t❤✃♣✳ ✺ ▲ê✐ ♥ã✐ ➤➬✉ ✸✳ ❚ỉ♥❣ q✉❛♥ ✈➭ ❝✃✉ tró❝ ❝đ❛ ❧✉❐♥ ✈➝♥ ◆❣♦➭✐ ♣❤➬♥ ▲ê✐ ♥ã✐ ➤➬✉✱ ❑Õt ❧✉❐♥ ✈➭ ❚➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦✱ ♥é✐ ❞✉♥❣ ❝đ❛ ợ trì tr ➜➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥✳ ◆é✐ ❞✉♥❣ ❝❤Ý♥❤ tr♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ➤➵✐ sè ▲✐❡✱ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥✱ ➤➵✐ sè ❝♦♥ ❈❛rt❛♥✱ ❜✐Ĩ✉ ❞✐Ơ♥ ❝đ❛ sl2 ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❤Ư ♥❣❤✐Ư♠✳ ❈ơ t❤Ĩ ♥é✐ ❞✉♥❣ ❝đ❛ ❝❤➢➡♥❣ ♥➭② ➤➢ỵ❝ ❝❤✐❛ t❤➭♥❤ ❝➳❝ t✐Õt s❛✉✿ ✶✳✶✳ ➜➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥✳ ✶✳✷✳ ❇✐Ĩ✉ ❞✐Ơ♥ ❝đ❛ ✶✳✸✳ ➜➵✐ sè ❝♦♥ ❈❛rt❛♥✳ ✶✳✹✳ ❉➵♥❣ ❝♦♠♣❛❝t✳ ✶✳✺✳ ▼ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❤Ư ♥❣❤✐Ư♠✳ ❈❤➢➡♥❣ ✷✿ ❝❤✐Ị✉ t❤✃♣✳ sl2 ❇✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ✈➭ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ❝ã ◆é✐ ❞✉♥❣ ❝❤Ý♥❤ ❝đ❛ ❝❤➢➡♥❣ ♥➭② ❧➭ tr×♥❤ ❜➭② ❧ý t❤✉②Õt ❝✃✉ tró❝ ❝đ❛ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥✱ ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝đ❛ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ t❤❡♦ trä♥❣ ✈➭ ✈❡❝t➡ trä♥❣✱ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ♥ö❛ ➤➡♥ ✈➭ ❝✉è✐ ❝ï♥❣ ❞ï♥❣ ♣❤➬♥ ♠Ị♠ trù❝ t✉②Õ♥ ▲✐❡ ❧✐Ưt ❦➟ ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝đ❛ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ❝ã sè ❝❤✐Ị✉ t❤✃♣✳ ◆é✐ ❞✉♥❣ ❝đ❛ ❝❤➢➡♥❣ ♥➭② ➤➢ỵ❝ ❝❤✐❛ t❤➭♥❤ ❝➳❝ t✐Õt s❛✉✿ ✷✳✶✳ ❚rä♥❣ ✈➭ ✈❡❝t➡ trä♥❣✳ ✷✳✷✳ ❙ù ❤♦➭♥ t♦➭♥ ❦❤➯ q✉②✳ ✷✳✸✳ P❤➞♥ ❧♦➵✐ ❝➳❝ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥✳ ✷✳✹✳ ❇✐Ĩ✉ ❞✐Ơ♥ trù❝ ❣✐❛♦ ✈➭ ➤è✐ ♥❣➱✉✳ ✷✳✺✳ ❇✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝đ❛ ➤➵✐ sè ▲✐❡ sl2 ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ ❚❤➬② ❣✐➳♦ ❚❙✳ ◆❣✉②Ơ♥ ◗✉è❝ ❚❤➡✳ ◆❤➞♥ ❞Þ♣ ♥➭②✱ t➳❝ ❣✐➯ ①✐♥ ợ tỏ ò í trọ ết ➤➲ t❐♥ t×♥❤ ❣✐ó♣ ➤ì t➠✐ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ✻ ▲ê✐ ♥ã✐ ➤➬✉ ❚➳❝ ❣✐➯ ①✐♥ ➤➢ỵ❝ ❝➯♠ ➡♥ ❝➳❝ ❚❤➬② ✭❈➠✮ tr♦♥❣ ❈❤✉②➟♥ ♥❣➭♥❤ ➜➵✐ sè ✈➭ ▲ý t❤✉②Õt sè✱ ❝➳❝ ❚❤➬② ✭❈➠✮ tr♦♥❣ ♥❣➭♥❤ ❚♦➳♥ ❝đ❛ ❱✐Ư♥ ❙➢ ♣❤➵♠ tù ♥❤✐➟♥✱ P❤ß♥❣ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝✱ ❇❛♥ ●✐➳♠ ❤✐Ư✉ ✈➭ ❝➳❝ P❤ß♥❣ ❜❛♥ ❝❤ø❝ ♥➝♥❣ ❝đ❛ ❚r➢ê♥❣ ➜❍ ❱✐♥❤ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ➤Ĩ t➳❝ ❣✐➯ ❤♦➭♥ t❤➭♥❤ ♥❤✐Ư♠ ✈ơ ❝đ❛ ♠ét ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝✳ ❚➳❝ ❣✐➯ ①✐♥ ➤➢ỵ❝ ❝➯♠ ➡♥ ❝➳❝ ❚❤➬② ✭❈➠✮✱ ❝➳❝ ➤å♥❣ ♥❣❤✐Ö♣ ♥➡✐ t➳❝ ❣✐➯ ➤❛♥❣ ❣✐➯♥❣ ❞➵② ✈➭ ❝➠♥❣ t➳❝ ➤➲ t➵♦ ➤✐Ò✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐✱ ❝ỉ ✈ị✱ ➤é♥❣ ✈✐➟♥ ✈➭ ❣✐ó♣ ➤ì t➳❝ ❣✐➯ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❧➭♠ ❧✉❐♥ ✈➝♥ tèt ♥❣❤✐Ư♣✳ ❈➯♠ ➡♥ sù ❤② s✐♥❤ ❝đ❛ ì ề t ỗ ự t t ữ ❝❤➽❝ ➤Ĩ t➳❝ ❣✐➯ ✈➢ỵt q✉❛ ❦❤ã ❦❤➝♥✱ ❤♦➭♥ t❤➭♥❤ ♥❤✐Ư♠ ✈ơ ❤ä❝ t❐♣ ❝đ❛ ♠×♥❤✳ ❳✐♥ tr➞♥ trä♥❣ ❦Ý♥❤ t➷♥❣ ●✐❛ ➤×♥❤ t❤➞♥ ②➟✉ ♠ã♥ q✉➭ t✐♥❤ t❤➬♥ ♥➭② ✈í✐ t✃♠ ❧ß♥❣ ❜✐Õt ➡♥ ❝❤➞♥ t❤➭♥❤ ♥❤✃t✳ ▼➷❝ ❞ï ➤➲ ❝ã ♥❤✐Ị✉ ❝è ❣➽♥❣ ♥❤➢♥❣ ❞♦ ♥➝♥❣ ❧ù❝ ❝ß♥ ♥❤✐Ị✉ ❤➵♥ ❝❤Õ✱ ♥➟♥ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ tế sót rt ợ ữ ❣ã♣ ý ❝đ❛ ❝➳❝ ♥❤➭ ❦❤♦❛ ❤ä❝ ✈➭ ➤å♥❣ ♥❣❤✐Ư♣ ➤Ĩ ❧✉❐♥ ✈➝♥ ❝ã t❤Ĩ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥ tèt ❤➡♥✳ ◆❣❤Ö ❆♥✱ ♥❣➭② ✺ t❤➳♥❣ ✼ ♥➝♠ ✷✵✶✼ ❚➳❝ ❣✐➯ ❚r➬♥ ❍➯✐ ◆❤➞♥ ❈❤➢➡♥❣ ✶ ➜➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ ◆é✐ ❞✉♥❣ tr♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❧➵✐ ♠ét ❝➳❝❤ ❝ã ❤Ö t❤è♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥✱ tỉ♥❣ q✉➳t ✈Ị ➤➵✐ sè ▲✐❡✱ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥✱ ➤➵✐ sè ❈❛rt❛♥✱ ❤Ö ♥❣❤✐Ö♠✱✳✳✳✳ ✶✳✶ ➜➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ✶✳✶✳✶✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ❣✐❛♥ ✈❡❝t➡ G ❈❤♦ K ❧➭ ♠ét tr➢ê♥❣ ✈➭ G ❧➭ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ tr➟♥ K ❑❤➠♥❣ ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ➤➵✐ sè ▲✐❡ tr➟♥ K ❤❛② K− ➤➵✐ sè ▲✐❡ ♥Õ✉ tr➟♥ G ➤➢ỵ❝ tr❛♥❣ ♠ét ♣❤Ð♣ ♥❤➞♥ ❣ä✐ ❧➭ tÝ❝❤ ▲✐❡ [., ]: G × G −→ G (X, Y ) −→ [X, Y ] s❛♦ ❝❤♦ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➞② t❤á❛ ♠➲♥✿ L1 ❚Ý❝❤ ▲✐❡ ❧➭ t♦➳♥ tö s♦♥❣ t✉②Õ♥ tÝ♥❤✱ tø❝ ❧➭✿ [λX + µY, Z] = λ[X, Z] + µ[Y, Z], [X, λY + µZ] = λ[X, Y ] + µ[X, Z], ∀X, Y, Z ∈ G, ∀λ, µ ∈ K L2 ❚Ý❝❤ ▲✐❡ ♣❤➯♥ ①ø♥❣✱ tø❝ ❧➭✿ [X, Y ] = −[Y, X], [X, X] = 0, ∀X, Y ∈ G L3 ❚Ý❝❤ ▲✐❡ t❤á❛ ♠➲♥ ➤➻♥❣ t❤ø❝ ❏❛❝➠❜✐✱ tø❝ ❧➭✿ [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0, X, Y, Z ∈ G ✼ ❈❤➢➡♥❣ ✶✳ ❙è ❝❤✐Ị✉ ❝đ❛ ➤➵✐ sè ▲✐❡ ➜➵✐ sè ▲✐❡ ❱Ý ❞ơ ✶✳ ✽ ➜➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ ❈❤♦ G G ❝❤Ý♥❤ ❧➭ sè ❝❤✐Ị✉ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ ➤➢ỵ❝ ❣ä✐ ❧➭ ❣✐❛♦ ❤♦➳♥ ♥Õ✉ G [X, Y ] = [Y, X], ∀X, Y ∈ G (A, ) ❧➭ ➤➵✐ sè ết ợ tr trờ K ị ĩ é t [−, −] : A × A −→ A, (x, y) −→ [x, y] = x.y − y.x, ∀(x, y) ∈ A × A ❑❤✐ ➤ã (A, [−, −]) trë t❤➭♥❤ ♠ét ➤➵✐ sè ▲✐❡ tr➟♥ tr➢ê♥❣ K ◆ã✐ r✐➟♥❣ t❛ ❝ã ➤➵✐ sè M at(n, K) ❝➳❝ ♠❛ tr❐♥ ✈✉➠♥❣ ❝✃♣ n ♣❤➬♥ tö tr➟♥ K ❧➭ ♠➠t ➤➵✐ sè tí ợ ị [A, B] = A.B − B.A, ∀A, B ∈ M at(n, K), tr♦♥❣ ➤ã ♣❤Ð♣ ♥❤➞♥ ✬✬✳✬✬ ❧➭ ♣❤Ð♣ ♥❤➞♥ ❝➳❝ ♠❛ tr❐♥✳ ➜➵✐ sè M at(n, K) ➤➢ỵ❝ ❦ý ❤✐Ư✉ ❧➭ gl(n, K) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ gl(n) ❱Ý ❞ơ ✷✳ ➤Þ♥❤✿ ❈➳❝ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ ❞➢í✐ ➤➞② ➤Ị✉ ❧➭ ❝➳❝ ➤➵✐ sè ▲✐❡ ✈í✐ tÝ❝❤ ▲✐❡ ➤➢ỵ❝ ①➳❝ [A, B] = AB − BA, ✈í✐ A, B ✭✶✮ ➜➵✐ sè ▲✐❡ t✉②Õ♥ tÝ♥❤ ➤➷❝ ❜✐Öt ✭✷✮ ➜➵✐ sè ▲✐❡ trù❝ ❣✐❛♦ ✭✸✮ ➜➵✐ sè ▲✐❡ ✉♥✐t❛r② ❈❤♦ V n sln = {A ∈ M at(n, K) | T r(A) = 0} o(n) = {A ∈ M at(n, K) | A + AT = 0} un = {A ∈ M at(n, K) | A + A∗ = 0} ✭✹✮ ➜➵✐ sè ▲✐❡ ✉♥✐t❛r② ➤➷❝ ❜✐Ưt ❱Ý ❞ơ ✸✳ ❧➭ ❝➳❝ ♠❛ tr❐♥ ✈✉➠♥❣ ❝✃♣ sun = {A ∈ un | T r(A) = 0} ❧➭ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ ❤÷✉ ❤➵♥ ❝❤✐Ị✉ tr➟♥ ❝➳❝ t♦➳♥ tö t✉②Õ♥ tÝ♥❤ tr➟♥ C ❳Ðt A = End(V ) ➤➵✐ sè C− ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ V ❑❤✐ ➤ã End(V ) trë t❤➭♥❤ ♠ét ➤➵✐ sè ▲✐❡✱ ✈í✐ tí ợ ị [f, g] = g.f f.g, ∀f, g ∈ End(V ), tr♦♥❣ ➤ã ♣❤Ð♣ ♥❤➞♥ ✬✬✳✬✬ ❧➭ ♣❤Ð♣ ❤ỵ♣ t❤➭♥❤ ❤❛✐ t♦➳♥ tư t✉②Õ♥ tÝ♥❤ tr➟♥ ♥➭② ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➵✐ sè ▲✐❡ t✉②Õ♥ tÝ♥❤ tỉ♥❣ q✉➳t✳ ❚❛ ✈✐Õt ❦❤✐ ❤✐Ĩ✉ A ➤å♥❣ ♥❤✃t gl(V ) ❧➭ ♠ét ➤➵✐ sè ▲✐❡✳ ◆Õ✉ t❛ ❝è ➤Þ♥❤ ♠ét ❝➡ së ❝đ❛ gl(V ) ✈í✐ gl(n, C) ➤➵✐ sè ❝➳❝ ♠❛ tr❐♥ ✈✉➠♥❣ ❝✃♣ n V t❤❛② ❝❤♦ V, ➜➵✐ sè End(V ) ❦❤✐ ➤ã t❛ ❝ã t❤Ó ♣❤➬♥ tư tr➟♥ C, ✈í✐ dimgl(V ) = dimgl(n, C) = n2 P❤Ð♣ ❤ỵ♣ t❤➭♥❤ ❤❛✐ t♦➳♥ tư t✉②Õ♥ tÝ♥❤ tr➟♥ V ♣❤Ð♣ ♥❤➞♥ ♠❛ tr❐♥ ❤❛✐ ♠❛ tr❐♥ t➢➡♥❣ ø♥❣✳ ❚❛ ①➳❝ ➤Þ♥❤ tÝ❝❤ ▲✐❡ tr➟♥ gl(n, C) : ❧➭ ✾ ➜➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ ❈❤➢➡♥❣ ✶✳ ét sở {eij } ủ gl(n, C), ỗ tr eij ợ ị s tử t➵✐ ✈Þ trÝ (i, j) ❜➺♥❣ 1, ❝➳❝ ♣❤➬♥ tư ò Pé tr ợ ①➳❝ ♥Õ✉ k = j ➤Þ♥❤ tr➟♥ ❝➡ së✿ eij ekl = δjk eil , ë ➤➞② δjk = ❧➭ ❦ý ❤✐Ö✉ ❑r♦♥❡❝❦❡r✳ ♥Õ✉ k = j ❉♦ tÝ♥❤ s♦♥❣ t✉②Õ♥ tÝ♥❤ ❝đ❛ tÝ❝❤ ▲✐❡ ♥➟♥ ➤Ĩ ①➳❝ ➤Þ♥❤ tÝ❝❤ ▲✐❡✱ t❛ ❝❤Ø ❝➬♥ ①➳❝ ➤Þ♥❤ tr➟♥ ❝➡ së {eij } ♥❤➢ s❛✉✿ eij , ekl = eij ekl − ekl eij = δjk eil − δli ekj , ➤➞② ❧➭ ♠❛ tr❐♥ ❝ã ❝➳❝ t❤➭♥❤ ♣❤➬♥ ❧➭ ❱Ý ❞ô ✹✳ ❈❤♦ 0, 1, −1 A ❧➭ ➤➵✐ sè tr➟♥ tr➢ê♥❣ K ✈➭ ϕ ∈ EndK (A) ❑❤✐ ➤ã t♦➳♥ tö t✉②Õ♥ tÝ♥❤ ϕ : A −→ A ➤➢ỵ❝ ❣ä✐ ❧➭ t♦➳♥ tư ✈✐ ♣❤➞♥ tr➟♥ A ♥Õ✉ ♥ã t❤á❛ ♠➲♥ ❝➠♥❣ t❤ø❝ ▲❡✐❜♥✐③✿ ϕ(x.y) = ϕ(x).y − x.ϕ(y) ❚❐♣ ❝➳❝ t♦➳♥ tư ✈✐ ♣❤➞♥ tr➟♥ A ➤➢ỵ❝ ❦ý ❤✐Ö✉ ❧➭ Der(A) ❑❤✐ ➤ã ∀ϕ, ϕ ∈ Der(A), t❛ ❝ã✿ [ϕ, ϕ ](a.b) = (ϕϕ − ϕ ϕ)(a.b) = ϕ(ϕ (a).b − a.ϕ (b)) − ϕ (ϕ(a).b − a.ϕ(b)) = (ϕϕ − ϕ ϕ)(a).b − a.(ϕϕ − ϕ ϕ)(b) = [ϕ, ϕ ](a).b − a.[ϕ, ϕ ](b) ❉♦ ➤ã [ϕ, ϕ ] ∈ Der(A) ❱❐② Der(A) ❧➭ ➤➵✐ sè ▲✐❡ ❝♦♥ ❝đ❛ gl(A) ✶✳✶✳✷✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ❣✐❛♥ ❝♦♥ ❈❤♦ G ❧➭ ➤➵✐ sè ▲✐❡ ✈➭ H ❧➭ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ ❝♦♥ ❝đ❛ G ❑❤➠♥❣ H ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➵✐ sè ▲✐❡ ❝♦♥ ❝đ❛ G, ♥Õ✉ H ➤ã♥❣ ✈í✐ tÝ❝❤ ▲✐❡✱ tø❝ ❧➭✿ ∀X, Y ∈ H t❤× [X, Y ] ∈ H ❱Ý ❞ô ✺✳ ❳Ðt ➤➵✐ sè ▲✐❡ gl(n) ❑ý ❤✐Ö✉ n sl(n) = A = [aij ]n ∈ gl(n) | T r(A) := aii = i=1 ❧➭ t❐♣ ❤ỵ♣ ❝➳❝ ♠❛ tr❐♥ ✈✉➠♥❣ ❝✃♣ n, ♣❤➬♥ tö ♣❤ø❝ ✈➭ ❝ã ✈Õt ❜➺♥❣ ❦❤➠♥❣✳ ❑❤✐ ➤ã sl(n) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ ❝♦♥ ❝đ❛ gl(n), ✈×✿ T r(aA + bB) = a.T r(A) + b.T r(B) = 0, ∀a, b ∈ C, ∀A, B ∈ sl(n, C) ❈❤➢➡♥❣ ✷✳ ❇✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ✈➭ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ❝ã ❝❤✐Ị✉ t❤✃♣ ✷ ✷✳✷✳✸✳ ❇ỉ ➤Ị✳ ✭①❡♠ ❬✹❪✮ ●ä✐ ❜✃t ❦❤➯ q✉②✮ ✈í✐ ❞➵♥❣ ✈Õt ϕ ❧➭ ❜✐Ĩ✉ ❞✐Ơ♥ ❝đ❛ G tr➟♥ V ✭❜✃t ❦❤➯ q✉② ❤♦➷❝ ❦❤➠♥❣ Tϕ (X, Y ) = T r(ϕ(X), ϕ(Y )) ◆Õ✉ ϕ ❧➭ ❜✐Ĩ✉ ❞✐Ơ♥ tr✉♥❣ t❤➭♥❤ t❤× ❞➵♥❣ ✈Õt tr➟♥ ❧➭ ❦❤➠♥❣ s✉② ❜✐Õ♥✳ ✷✳✷✳✹✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❚♦➳♥ tư ❈❛s✐♠✐r✮✳ ❈❤♦ I ❧➭ ♠ét ✐➤➟❛♥ ❞✉② ♥❤✃t ❝đ❛ G ❜ï ✈í✐ kerϕ ✈➭ sù ❤➵♥ ❝❤Õ ϕ ①➳❝ ➤Þ♥❤ ♠ét ❜✐Ĩ✉ ❞✐Ơ♥ tr✉♥❣ t❤➭♥❤ ❝đ❛ I ●ä✐ X1 , X2 , , Xn ❧➭ ♠ét ❝➡ së tï② ý ❝ñ❛ I s❛♦ ❝❤♦ I ✈➭ Y1 , Y2 , , Yn ❝đ❛ Γϕ ❣✐❛♦ ❤♦➳♥ ✈í✐ ♠ä✐ t♦➳♥ tư ϕ(X) T r(Γϕ ) = dim I = dim G − dim kerϕ ❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ❜✃t ❦ú X ∈ G ❚❛ ❦❤❛✐ tr✐Ó♥ [X, Xi ] = ❚❛ ❝ã ❧➭ t♦➳♥ ϕ ✷✳✷✳✺✳ ▼Ư♥❤ ➤Ị✳ ✭✐✮ ❚♦➳♥ tư ❈❛s✐♠✐r ✭✐✐✮ ϕ(Xi ) ◦ ϕ(Yi ) ❑❤✐ ➤ã t❛ ❣ä✐ Γϕ Tϕ (X, Y ) = δij ➜➷t Γϕ = tư ❈❛s✐♠✐r ❧➭ ♠ét ❝➡ së ➤è✐ ♥❣➱✉ ✈í✐ ❞➵♥❣ ✈Õt tr➟♥ xij = T r[X, Xi ]Yj xij Xj , ✈➭ [X, Yi ] = −T rXi [X, Yi ] = −yij , yij Yj ❜ë✐ ✈× tÝ♥❤ ❜✃t ❜✐Õ♥ ❝ñ❛ Tϕ ❑❤✐ ➤ã [X, Γϕ ] = ❱❐② t❛ ❝ã [X., Xi ]Yi + Xi [X, Yi ] = xij Xj Yi + yij Xi Yj = (i) ❑❤➻♥❣ ➤Þ♥❤ (ii) s✉② r❛ tõ sù ❦✐Ö♥ T rXi Yi = ✷✳✷✳✻✳ ❍Ö q✉➯✳ ◆Õ✉ ϕ ❜✃t ❦❤➯ q✉② ✈➭ V = {0} t❤× Γϕ ❧➭ t♦➳♥ tư ✈➠ ❤➢í♥❣ dim G − dim kerϕ id dim V ❉♦ ➤ã✱ Γϕ ❦❤➠♥❣ s✉② ❜✐Õ♥ ♥Õ✉ ✷✳✷✳✼✳ ▼Ư♥❤ ➤Ị✳ ❈❤♦ ϕ ❧➭ ❦❤➠♥❣ t➬♠ t❤➢ê♥❣✳ G t➳❝ ➤é♥❣ tr➟♥ V v∈G s❛♦ ❝❤♦ ✷✳✷✳✽✳ ➜Þ♥❤ ❧ý✳ ▼ä✐ ❜✐Ĩ✉ ❞✐Ơ♥ ❝đ❛ ✷✳✷✳✾✳ ➜Þ♥❤ ❧ý✳ ❈❤♦ G ϕ1 ✈➭ ϕ2 ➤Ị✉ ❤♦➭♥ t♦➭♥ ❦❤➯ q✉②✳ ϕ ❝đ❛ G ❝ñ❛ G1 ❧➭ f (X) = Xv, ∀X ∈ G ❧➭ tỉ♥❣ trù❝ t✐Õ♣ ❝đ❛ ❤❛✐ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ❜✃t ❦ú ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② G f : G −→ V f ([X, Y ]) = Xf (X)−Y f (X), ∀X, Y ∈ G ♠ét ❤➭♠ t✉②Õ♥ tÝ♥❤ t❤á❛ ♠➲♥ q✉❛♥ ❤Ö ❑❤✐ ➤ã✱ tå♥ t➵✐ ♠ét ✈❡❝t➡ ♥❤➢ ➤➲ ➤Þ♥❤ ♥❣❤Ü❛ ë tr➟♥✳ ●ä✐ ✈➭ G1 ✈➭ G2 ❑❤✐ ➤ã✱ ➤Ị✉ ♣❤➞♥ tÝ❝❤ t❤➭♥❤ tÝ❝❤ t❡♥s♦r ❝đ❛ ❤❛✐ ❜✐Ĩ✉ G2 ✸✻ ❈❤➢➡♥❣ ✷✳ ✷✳✸ ❈❤♦ ❇✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ✈➭ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ ❝ã ❝❤✐Ị✉ t❤✃♣ ✷ P❤➞♥ ❧♦➵✐ ❝➳❝ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ R ❧➭ ♠ét ❤Ö ♥❣❤✐Ö♠ tr♦♥❣ ♠ét ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ ♣❤ø❝ ❦❤➠♥❣ ❣✐❛♥ ➤è✐ ♥❣➱✉ ❝đ❛ së ➤è✐ ✈í✐ ✈➭ G s❛♦ ❝❤♦ ❑ý ❤✐Ö✉ H ❧➭ V = H∗ ●ä✐ S = {α1 , α2 , , αl } ❧➭ ♠ét ❝➡ R, ❣ä✐ H1 , H2 , , Hl ∈ H ❧➭ ❝➳❝ ♥❣❤✐Ư♠ ♥❣❤Þ❝❤ ➤➯♦ ❝đ❛ α1 , α2 , , αl aij = αi , Hi ●ä✐ V V ❧➭ ❝➳❝ sè ♥❣✉②➟♥ ❈❛rt❛♥✳ ❧➭ ♠ét ➤➵✐ sè ▲✐❡ ①➳❝ ➤Þ♥❤ ❜ë✐ ✭t➢➡♥❣ ø♥❣ ✈í✐ ❝➳❝ ♣❤➬♥ tö 3l ♣❤➬♥ tö s✐♥❤ ei , fi , hi ✈í✐ 1≤i≤l Xi , X−i , Hi ) ✈➭ ❝➳❝ q✉❛♥ ❤Ö ✭✶✮ [hi , hj ] = 0, ✭✷✮ [hi , ej ] = aji ej ✭✸✮ ✈➭ [hi , fj ] = −aji fj , [ei , fj ] = 0, ✭✹✮ [ei , [ei , [ [ei , ej ] ]]] = ✈í✐ − aji + ♥❤➞♥ tư ei ✭✺✮ [fi , [fi , [ [fi , fj ] ]]] = ✈í✐ − aji + ♥❤➞♥ tư fi ❚❛ ❝ã t❤Ĩ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ ❤Ư ♥❣❤✐Ư♠ ❝đ❛ ♥ã ❧➭ R, ❝➳❝ hi G ❧➭ ♠ét ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ✭❤÷✉ ❤➵♥ ❝❤✐Ị✉✮ ✈í✐ ❤×♥❤ t❤➭♥❤ ♠ét ➤➵✐ sè ▲✐❡ ❝♦♥ ❈❛rt❛♥ H ▼ét ❧ý t❤✉②Õt ❝✃✉ tró❝ ♥❣❤✐Ư♠ ❝❤♦ ♣❤Ð♣ t❛ ♣❤➞♥ ❧♦➵✐ t✃t ❝➯ ➤➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ tr➟♥ tr➢ê♥❣ ➤ã♥❣ ➤➵✐ sè ✈➭ ♠ét tr➢ê♥❣ ❦❤➠♥❣ ➤ã♥❣ ➤➵✐ sè✳ ❙❛✉ ➤➞② t❛ t✐Õ♥ ❤➭♥❤ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ tr➟♥ tr➢ê♥❣ sè ♣❤ø❝ ❚r➟♥ tr➢ê♥❣ ❝➡ së ❧➭ C C, t❤× ❝ã ✹ ❞➲② ✈➠ ❤➵♥ ✧❝ỉ ➤✐Ĩ♥✧ ➤ã ❧➭✿ An (n ≥ 1), Bn (n ≥ 2), Cn (n ≥ 3), Dn (n ≥ 4) En (n = 6, 7, 8), F4 ❝ñ❛ ✈➭ G2 ✈➭ ✺ ➤➵✐ sè ▲✐❡ ➤➡♥ ✧♥❣♦➵✐ ❧Ö✧✿ ❚✃t ❝➯ ♥❤÷♥❣ ➤➵✐ sè ▲✐❡ ♥➭② ➤Ị✉ ❧➭ ➤➵✐ sè ▲✐❡ ❝♦♥ gl(n, C) ✈í✐ n t❤Ý❝❤ ❤ỵ♣✱ ❝ã ♥❣❤Ü❛ ❧➭ ♥❤÷♥❣ ♣❤➬♥ tư ♥➭② ❧➭ ❝➳❝ ♠❛ tr❐♥ ❝ã ❦Ý❝❤ t❤➢í❝ ♣❤ï ❤ỵ♣✳ ❚❛ ✈✐Õt Eij ➤Ĩ ❝❤Ø ♠❛ tr ị ị trí ij ò ë ❝➳❝ ✈Þ trÝ ❦❤➳❝✳ ❚❛ sư ❞ơ♥❣ ❝➳❝ ✈❡❝t➡ ❝➡ së ❝❤✉➮♥ ❝ñ❛ C ✈➭ ❝➳❝ ❤➭♠ t✉②Õ♥ tí i r ỗ trờ ợ t ❤✐Ư♥ t❤Þ ♠ét ➤➵✐ sè ▲✐❡ ❝♦♥ ❣✐❛♦ ❤♦➳♥ H ♠➭ t❤ù❝ tÕ ♥ã ❧➭ ♠ét ❈❙❆ ✈➭ ❝➳❝ ♥❣❤✐Ö♠ t➢➡♥❣ ø♥❣✱ ❤Ư ❝➡ ❜➯♥ ✈➭ ❝➳❝ ♣❤➬♥ tư ♥❣❤✐Ư♠✱ ✈➭ ❝ị♥❣ ❧➭ ❝➳❝ ➤è✐ ♥❣❤✐Ư♠ ❝➡ ❜➯♥ ✈➭ ♠❛ tr❐♥ ❈❛rt❛♥✳ ◆Õ✉ ϕ ❧➭ ♠ét ❜✐Ĩ✉ ❞✐Ơ♥ ❝đ❛ G tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ ➤Ĩ ❝❤Ø ❜✐Ĩ✉ ❞✐Ơ♥ ❝➯♠ s✐♥❤ tr➟♥ tÝ❝❤ ♥❣♦➭✐ V V t❛ ✈✐Õt ϕ ∧ ϕ ❤♦➷❝ ✈➭ ♠ét ❝➳❝❤ tæ♥❣ q✉➳t ❤➡♥ r ϕ ϕ ➤Ĩ ✸✼ ❈❤➢➡♥❣ ✷✳ ❇✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ✈➭ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ ❝ã ❝❤✐Ị✉ t❤✃♣ ✷ ❝❤Ø ❜✐Ĩ✉ ❞✐Ơ♥ ❝➯♠ s✐♥❤ tr➟♥ ❧ị② t❤õ❛ t❤ø t✐Õt ❤➡♥✿ ◆Õ✉ i ♥➭♦✳ ❑❤✐ ➤ã✱ ♥Õ✉ ϕ ❧➭ trù❝ ❣✐❛♦ ✭t➢➡♥❣ ø♥❣ ➤è✐ ♥❣➱✉✮ ❚r➢í❝ ❤Õt t❛ t❤✃②✱ tr➟♥ ϕ ❑❤✐ ➤ã✱ ♠ét ➤➻♥❣ ❝✃✉ ➤➻♥❣ ❜✐Õ♥ b1 : V −→ ϕ ±b Vi tr➟♥ ❝ò♥❣ ✈❐②✳ ❈❤ø♥❣ ♠✐♥❤✳ ϕ G ❜ë✐ ❣✐➯ t❤✉②Õt λ b : V i tr ỗ s r ♠ét ➳♥❤ ①➵ t➢➡♥❣ tù ϕ1 − tù ➤è✐ ♥❣➱✉✳ ◆Õ✉ ➤è✐ ♥❣➱✉ ❝ñ❛ b ❧➭ ±b1 ❚❛ ❝ã ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ t❤× ➤è✐ ♥❣➱✉ ❝đ❛ b1 ❧➭ ✷✳✹✳✻✳ ❇ỉ ➤Ị✳ ❱í✐ ϕ1 ❧➭ tỉ♥❣ trù❝ t✐Õ♣ ❝đ❛ ❝➳❝ V ❧➭ trä♥❣ tré✐ ❝❤♦ tr➢í❝✳ ❇✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ϕλ ❧✐➟♥ ❦Õt ✈í✐ λ ❧➭ tù ➤è✐ ♥❣➱✉ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ trä♥❣ ♥❤ë ♥❤✃t ❝ñ❛ ♥ã ❧➭ −λ ◆ã✐ ❝➳❝❤ ❦❤➳❝✱ ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ϕλ ❧➭ tù ➤è✐ ♥❣➱✉ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ sù ➤è✐ ❧❐♣ ❜✐Õ♥ ϕ (X) = −ϕ(X) ❈❤ø♥❣ ♠✐♥❤✳ ❚õ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ➤è✐ ♥❣➱✉ trä♥❣ ❝đ❛ ➤Ị✉ ❧➭ ♣❤đ ➤Þ♥❤ ❝➳❝ trä♥❣ ❝đ❛ ➤➵✐ ❝đ❛ ϕ φλ ϕλ ❉♦ ➤ã✱ λ ❧➭ ♣❤đ ➤Þ♥❤ ❝➳❝ trä♥❣ ❝ù❝ t✐Ĩ✉ ❝ñ❛ λ t❤➭♥❤ −λ ❝❤ø♥❣ tá r➺♥❣ ❝➳❝ ❧➭ trä♥❣ ❝❛♦ ♥❤✃t ✈➭ ❝ù❝ φλ ❚r♦♥❣ ✷✳✸ ❝❤ó♥❣ t❛ ➤➲ ♣❤➞♥ ❧♦➵✐ ❝➳❝ ➤➵✐ sè ▲✐❡ ➤➡♥ ✧❝ỉ ➤✐Ĩ♥✧✿ Bn (n ≥ 2), Cn (n ≥ 3) ✈➭ Dn (n ≥ 4) An (n ≥ 1), ◆é✐ ❞✉♥❣ ❝❤Ý♥❤ ❝đ❛ ➤Þ♥❤ ❧ý s❛✉ ➤➞② ❝❤♦ ♣❤Ð♣ t❛ ♠➠ t➯ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❝đ❛ ♥ã✳ ✷✳✹✳✼✳ ➜Þ♥❤ ❧ý✳ ✭✐✮ ➜è✐ ✈í✐ An : ϕλ ❧➭ tù ➤è✐ ♥❣➱✉ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ f1 = f2 + fn = f3 + fn−1 = · · · ✭♠ét ❝➳❝❤ t➢➡♥❣ ➤➢➡♥❣ ◆ã ❧➭ ➤è✐ ♥❣➱✉ ♥Õ✉ m1 = mn , m2 = mn−1 , ) n ≡ ♠♦❞ f1 trự tr trờ ợ ò ❧➵✐✳ ✹✸ ❈❤➢➡♥❣ ✷✳ ❇✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ✈➭ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ❝ã ❝❤✐Ị✉ t❤✃♣ ✷ ✭✐✐✮ ➜è✐ ✈í✐ Bn : ϕλ ❧✉➠♥ ❧➭ tù ➤è✐ ♥❣➱✉✳ ◆ã ❧➭ ➤è✐ ♥❣➱✉ ♥Õ✉ n ≡ ♠♦❞ ❤♦➷❝ n ≡ ♠♦❞ ✈➭ ❝➳❝ fi ➤Ị✉ ❧➭ ♥ư❛ ♥❣✉②➟♥ ✭✈í✐ mn ❧➭ ❧❰✮ ✈➭ trự tr trờ ợ ò ố Cn : ϕ λ m1 + m3 + m5 + · · · ✭✐✈✮ ➜è✐ ✈í✐ ❧✉➠♥ ❧➭ tù ➤è✐ ♥❣➱✉✳ ◆ã ❧➭ ➤è✐ ♥❣➱✉ ♥Õ✉ D n : ϕλ mn1 + mn ❧➭ tù ➤è✐ ♥❣➱✉ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ❤♦➷❝ n ❝❤➼♥ ❤♦➷❝ n ≡ ♠♦❞ ✈➭ ❝➳❝ fi n ❧❰ ✈➭ ➤Ị✉ ❧➭ ♥ư❛ ❧❰ trự tr trờ ợ ò ❝❤✐Ị✉✮ ❈❤♦ G ❧➭ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ n ❱× ❝➳❝ ♥❣❤✐Ư♠ ❝➡ ❜➯♥ α1 , α2 , αn ❧➭ ♠ét ❝➡ së ❝ñ❛ H0 ♣❤ø❝ ❝ã ❤➵♥❣ ❜➺♥❣ t➵✐ ♠ét ♣❤➬♥ tö ◆ã ❧➭ ➤è✐ ♥❣➱✉ ♥Õ✉ ✭➤➵✐ sè ▲✐❡ ❝♦♥ ❝❤Ý♥❤ ✷✳✹✳✽✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ❧➭ ❧❰ ✭tø❝ ❧➭ ❧➭ ❧❰ ✈➭ trù❝ ❣✐❛♦ tr♦♥❣ trờ ợ ò f1 = ế mn1 = mn ✮✳ ♥❣✉②➟♥ ♥Õ✉ fi Hp ∈ H0 s❛♦ ❝❤♦ αi (Hp ) = 2, ≤ i ≤ n ❚❛ ✈✐Õt Hp = ❝❤ä♥ ❝➳❝ ❤➭♥❣ sè ci , c−i s❛♦ ❝❤♦ ci c−i = pi ✈➭ Xp = C − iHi , X−p = tå♥ pi Hi , C−i Hi ❙ư ❞ơ♥❣ ❝➳❝ q✉❛♥ ❤Ư [H, Xi ] = αi (H)X−i , [Xi , X−i ] = Hi , [Xi , X−j ] = ✈í✐ i=j t❛ t❤ư ❧➵✐ ➜➵✐ sè ▲✐❡ ❝♦♥ ❝đ❛ ➤ã [Hp , Xp ] = 2Xp , [Hp , X−p ] = −2Xp 2X−p , [Xp , X−p ] = Hp G ❧➭ Gp s✐♥❤ r❛ ❜ë✐ Hp , Xp , X−p ❤✐Ó♥ ♥❤✐➟♥ ❝ï♥❣ ❧♦➵✐ A1 ❑❤✐ Gp ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➵✐ sè ▲✐❡ ❝♦♥ ❝❤Ý♥❤ 3− ❝❤✐Ị✉✱ ✈✐Õt t➽t ❧➭ P T D ❚õ ➤Þ♥❤ ♥❣❤Ü❛ t❛ t❤✃② P T D ❝ã ➤➵✐ sè ❝♦♥ ❈❛rt❛♥ ❧➭ CHp ❍Ö ♥❣❤✐Ö♠ ❜❛♦ ❣å♠ ❝➳❝ ±αp ①➳❝ ➤Þ♥❤ ❜ë✐ αp (Hp ) = ▼ét ❜✐Ĩ✉ ❞✐Ơ♥ tr♦♥❣ ϕ ❝đ❛ Gp ❧➭ ❤➵♥ ❝❤Õ ❝đ❛ ❜✐Ĩ✉ ❞✐Ơ♥ ϕ ❝đ❛ G ❱× Hp ∈ H ♥➟♥ ♠ét ρ ❝ñ❛ ϕ ❧➭ ❤➵♥ ❝❤Õ ❝ñ❛ trä♥❣ ρ ❝ñ❛ ϕ ✈➭ t✃t ❝➯ ❝➳❝ trä♥❣ ϕ ①✉✃t ❤✐Ö♥ t❤❡♦ ❝➳❝❤ ♥➭②✳ ◆ã✐ ❝❤✉♥❣✱ ϕ sÏ ❦❤➠♥❣ ❜✃t ❦❤➯ q✉② ✈➭ t❛ sÏ t➳❝❤ t❤➭♥❤ ♠ét tỉ♥❣ ❝đ❛ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝đ❛ ✷✳✹✳✾✳ ❇ỉ ➤Ị✳ ❈❤♦ Gp , tø❝ ❧➭ t➳❝❤ t❤➭♥❤ ❝➳❝ Ds ϕ = ϕλ ❧➭ ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝đ❛ G ✈í✐ trä♥❣ ❝❛♦ ♥❤✃t ❧➭ λ ❑❤✐ ➤ã ✭✐✮ λ ❧➭ trä♥❣ ❝ù❝ ➤➵✐ ❝ñ❛ ϕ ✈➭ ❝ã ❜é✐ ❧➭ ✭✐✐✮ ❚r♦♥❣ sù t➳❝❤ ϕ t❤➭♥❤ ❜✐Ĩ✉ ❞✐Ơ♥ Ds ✈í✐ 2s = λ(Hp ) ①➯② r❛ ➤ó♥❣ ♠ét ❧➬♥✳ ✹✹ ❇✐Ĩ✉ ❞✐Ô♥ ❜✃t ❦❤➯ q✉② ✈➭ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ❝ã ❝❤✐Ị✉ t❤✃♣ ✷ ❈❤➢➡♥❣ ✷✳ ❈❤ø♥❣ ♠✐♥❤✳ (i) ❈➳❝ trä♥❣ ❝đ❛ ✈í✐ ❝➳❝ sè ♥❣✉②➟♥ ❦❤➠♥❣ ➞♠ ρ(Hp ) = λ(Hp ) − s✉② r❛ ϕ ki ❦❤➳❝ ✈í✐ ❝❤Ý♥❤ ✈➭ λ ki > k − i < λ(Hp ) ❉♦ ➤ã✱ tõ ▼➷t ❦❤➳❝ λ αp (Hp ) = t❛ ❝ã tr♦♥❣ ϕ ♥➟♥ (i) ❝❤Ý♥❤ ①➳❝ ❧➭ H ✭✐✮ ϕλ ✭✐✐✮ ϕλ ❧➭ tù ➤è✐ ♥❣➱✉✳ ❑❤✐ ➤ã✿ ❧➭ trù❝ ❣✐❛♦ ♥Õ✉ ϕλ λ(Hp ) ❧❰✳ ❧➭ ➤è✐ ♥❣➱✉ ♥Õ✉ ❈❤ø♥❣ ♠✐♥❤✳ ❘â r➭♥❣ ϕλ λ(Hp ) ❝❤➼♥✳ ❧➭ trù❝ ❣✐❛♦ ♥Õ✉ ϕλ ❧➭ trù❝ ❣✐❛♦ ✈➭ ❧➭ ➤è✐ ♥❣➱✉✳ ❚❛ ➳♣ ❞ơ♥❣ ❦Õt q✉➯ ❝đ❛ ❇ỉ ➤Ị ✷✳✹✳✺ ➤è✐ ✈í✐ tr♦♥❣ ❝➳❝ ϕλ ϕλ ❧➭ ➤è✐ ♥❣➱✉ ♥Õ✉ ϕλ ✈➭ sù ♣❤➞♥ tÝ❝❤ ❝đ❛ ♥ã Ds ❱× t❤➭♥❤ ♣❤➬♥ ➤➬✉ ①➯② r❛ ❝❤Ø ♠ét ❧➬♥ t❤❡♦ ❇ỉ ➤Ị ✷✳✹✳✾✱ ❞♦ ➤ã tõ ❇ỉ ➤Ị ✷✳✹✳✺ t❛ t❤✃② t❤➭♥❤ ♣❤➬♥ ➤➬✉ ♥➭② ❧➭ trù❝ ❣✐❛♦ ♥Õ✉ ϕλ ❣✐➳ trÞ r✐➟♥❣ ❧í♥ ♥❤✃t 2s ✷✳✹✳✶✵✳ ❇ỉ ➤Ị✳ ●✐➯ sư ♥Õ✉ ki αi ❝ã ❜é✐ ❧➭ (ii) ❑Õt q✉➯ (ii) ❧➭ ♠ét ❤Ư q✉➯ ❝đ❛ (i) ✈× tr♦♥❣ ❜✃t ❦ú Ds ❝đ❛ ρ = λ− ➤Ị✉ ❝ã ❞➵♥❣ ❝ị♥❣ ❧➭ ➤è✐ ♥❣➱✉✳ ▼➷t ❦❤➳❝ ✷✳✹✳✶✶✳ ➜Þ♥❤ ❧ý✳ ϕλ ϕλ ❝ị♥❣ ❧➭ trù❝ ❣✐❛♦ ✈➭ ➤è✐ ♥❣➱✉ ❝❤➼♥ ✈➭ ➤è✐ ♥❣➱✉ ♥Õ✉ ✭①❡♠ ❬✺❪✮ ▼ét ♥❤ã♠ ❝♦♥ ❝♦♠♣❛❝t G ❝ñ❛ λ(Hp ) ❝❤➼♥✳ GL(n, C) ❧➭ ❧✐➟♥ ❤ỵ♣ GL(n, C) ✈í✐ ♠ét ♥❤ã♠ ❝♦♥ ❝ñ❛ ♥❤ã♠ trù❝ ❣✐❛♦ t❤ù❝ O(n) ✭♥❤ã♠ ➤è✐ ♥❣➱✉ n ✈í✐ n ❝❤➼♥✮ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ♥ã ➤Ĩ ❧➵✐ ❜✃t ❜✐Õ♥ ♠ét ❞➵♥❣ s♦♥❣ t✉②Õ♥ tÝ♥❤ ✉♥✐t❛ Sp n ➤è✐ ①ø♥❣ ✭➤è✐ ①ø♥❣ ❧Ö❝❤✮ ❦❤➠♥❣ s✉② ❜✐Õ♥ tr➟♥ C tr♦♥❣ ❚õ ❦Õt q✉➯ ❝đ❛ ➤Þ♥❤ ❧ý tr➟♥ t❛ t❤✃②✿ ❚✃t ❝➯ ❝➳❝ ❜✐Ĩ✉ ➤✐Ơ♥ ❝đ❛ Spin(n) ✈í✐ n ≡ ±1 ♠♦❞ ❤♦➷❝ n ≡ ±0 ♠♦❞ ❝đ❛ SO(n) ✈í✐ n ≡ ♠♦❞ ✈➭ ❝ñ❛ ❝➳❝ ♥❤ã♠ ❝♦♠♣❛❝t G2 , F4 , E8 ❝ã t❤Ĩ ❜✐Õ♥ ➤ỉ✐ t❤➭♥❤ ❞➵♥❣ ❝♦♠♣❛❝t✳ ❚❛ ❝❤ó ý✿ ❇✐Ĩ✉ ❞✐Ơ♥ ①♦➽♥ ♠♦❞ ∆±n ✈➭ ∆n ❝đ❛ Dn ❧➭ ➤è✐ ♥❣➱✉ ❝đ❛ Bn ❧➭ trù❝ ❣✐❛♦ ✈í✐ n ≡ ♠♦❞ ❤♦➷❝ n ≡ n ≡ ♠♦❞ ❤♦➷❝ n ≡ ♠♦❞ ❇✐Ó✉ ❞✐Ơ♥ ♥ư❛ ①♦➽♥ ❝đ❛ ❧➭ trù❝ ❣✐❛♦ ✈í✐ ❦❤➠♥❣ tù ➤è✐ ♥❣➱✉ ✈í✐ ∆n n ❧❰✳ n ≡ ♠♦❞ ✈➭ ∆±n ❧➭ ➤è✐ ♥❣➱✉ ✈í✐ n ≡ ♠♦❞ ✈➭ ✹✺ ❈❤➢➡♥❣ ✷✳ ✷✳✺ ❇✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ✈➭ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ ❝ã ❝❤✐Ị✉ t❤✃♣ ✷ ❇✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝đ❛ sl2 ◆é✐ ❞✉♥❣ ❝❤Ý♥❤ tr♦♥❣ t✐Õt ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❞❛♥❤ s➳❝❤ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② V, ✈í✐ dim V = d ❝đ❛ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥✱ ❝ï♥❣ ✈í✐ sù ♣❤➞♥ tÝ❝❤ ❜✐Ĩ✉ ❞✐Ơ♥ Λ2 V ❈❤ó♥❣ t➠✐ ❞ï♥❣ ♣❤➬♥ ♠Ị♠ trù❝ t✉②Õ♥ ▲✐❊ ✭①❡♠ ❬✸❪✮ ➤Ĩ tÝ♥❤ t♦➳♥ ❝➳❝ ❦Õt q✉➯✳ ❈➳❝ ❦ý ❤✐Ư✉ sù ❞ơ♥❣ tr♦♥❣ ❝❤➢➡♥❣ tr×♥❤ ❣✐è♥❣ ♥❤➢ ❝➳❝ ❦ý ❤✐Ư✉ ➤➲ tr×♥❤ ❜➭② ë tr➟♥ ❈➳❝❤ ❧➭♠ ♥❤➢ s❛✉✿ ❙❛✉ ❦❤✐ ➤➲ ❝❤ä♥ ➤➢ỵ❝ d ♥❣❤✐Ư♠ ➤➡♥ α1 , α2 , , αd , ❜✐Ĩ✉ ➤å ❉②♥❦✐♥ ➤➢ỵ❝ ❧❐♣ ✈í✐ t➢ ❝➳❝❤ ❣➳♥ ♥❤➲♥ tù ♥❤✐➟♥ ❝➳❝ ➤Ø♥❤ ❜ë✐ ❝➳❝ ♥❣❤✐Ö♠ ♥➭② ✈➭ ❝➳❝ trä♥❣ ❝➡ ❜➯♥ ω1 , ω2 , , ωd sÏ ❤♦➭♥ t ợ ị ó t ý ệ ể ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② t➢➡♥❣ ø♥❣ ✈í✐ ❝➳❝ trä♥❣ n1 ω1 +n2 ω2 +· · ·+nd ωd ❧➭ [n1 , n2 , , nd ωd ] ➜➷❝ ❜✐Ưt✱ [0, 0, , 0] ❧➭ ➤Ĩ ❝❤Ø ❜✐Ĩ✉ ❞✐Ơ♥ t➬♠ t❤➢ê♥❣ 1− ❝❤✐Ị✉✳ ❚r♦♥❣ ❦❤✉➠♥ ❦❤ỉ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❝❤ó♥❣ t➠✐ tÝ♥❤ t♦➳♥ t➢ê♥❣ ♠✐♥❤ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝❤♦ ➤➵✐ sè ▲✐❡ sl2 A1 = sl2 ◆❤➢ t❛ ➤➲ ❜✐Õt ❝ã ❤❛✐ ♥❤ã♠ ▲✐❡ ❧✐➟♥ t❤➠♥❣ ♥❤❐♥ ❧➭♠ ➤➵✐ sè ❝ñ❛ ❝❤ó♥❣✱ ➤ã ❧➭ ❱í✐ ♥❤ã♠ ▲✐❡ t❤ø ♥❤✃t✱ ❞✐Ơ♥ tù ♥❤✐➟♥ ❝đ❛ sl2 ❜✐Ĩ✉ ❞✐Ơ♥ tù ♥❤✐➟♥ SL2 SL2 ✈➭ P SL2 ❜✐Ĩ✉ ❞✐Ơ♥ tù ♥❤✐➟♥ W ❝đ❛ ♥ã ❝➯♠ s✐♥❤ ♠ét ❜✐Ó✉ ✈➭ t❛ ❜✐Õt r➺♥❣ t✃t ❝➯ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❦❤➳❝ ❝đ❛ sl2 ➤Ị✉ s✐♥❤ r❛ tù ♥➭② q✉❛ ❝➳❝ ♣❤Ð♣ t♦➳♥ ❝ñ❛ ➤➵✐ sè t✉②Õ♥ tÝ♥❤ ♥❤➢ tÝ❝❤ t❡♥s♦r✱ W tÝ❝❤ ♥❣♦➭✐✱ ❤➭♠ tö ❍♦♠✱ ♣❤Ð♣ ❧✃② ♥❣➱✉ ♥❤✐➟♥✱ ➜è✐ ✈í✐ ♥❤ã♠ ▲✐❡ P SL2 ✳✳✳ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝đ❛ ♥ã ❧➭ {[2n] | n ≥ 0} ❈➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ♥➭② ❝➯♠ s✐♥❤ ♠ét ❝➳❝❤ tù ♥❤✐➟♥ ♥➟♥ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❝đ❛ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❦❤➳❝ ❝đ❛ sl2 sl2 ✈➭ t✃t ❝➯ ➤Ị✉ ➤➢ỵ❝ s✐♥❤ r❛ tõ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❞➵♥❣ ♥➭② q✉❛ ❝➳❝ ♣❤Ð♣ t♦➳♥ ❝ñ❛ ➤➵✐ sè t✉②Õ♥ tÝ♥❤ ♥ã✐ tr➟♥✳ ❈❤➻♥❣ ❤➵♥✱ t❛ ❝ã t❤Ĩ t❤✉ ➤➢ỵ❝ ❜✐Ĩ✉ ❞✐Ơ♥ ✈❐②✱ t❛ ❝ã [2] tõ ❜✐Ĩ✉ ❞✐Ơ♥ [2n] ✈í✐ n > ❚❤❐t n Λ2 [2n] = ⊕ [4k − 2] k=1 ❚➢➡♥❣ tù ♥❤➢ tr➟♥✱ t❛ ①Ðt ❜✐Ĩ✉ ❞✐Ơ♥ [2n + 1] ✈í✐ n > ❑❤✐ ➤ã t❛ ❝ã n+1 sym2 [2n + 1] = ⊕ [4k − 2] ✈➭ ❞♦ ➤ã t❛ t❤✉ ➤➢ỵ❝ ❜✐Ĩ✉ ❞✐Ơ♥ [2] k=1 ❍➡♥ ♥÷❛✱ tõ ❞✐Ơ♥ [2] ⊗ [2k + 1] = [2k − 1] ⊕ [2k + 1] ⊕ [2n + 3] t❛ t❤✉ ➤➢ỵ❝ ❜✐Ĩ✉ [2k − 1] ❚❤❡♦ ❝➳❝❤ ♥➭②✱ ❜✐Ĩ✉ ❞✐Ơ♥ ♣❤Ð♣ t♦➳♥ ❝đ❛ ➤➵✐ sè t✉②Õ♥ tÝ♥❤✳ [1] ❝ã ➤➢ỵ❝ tõ ❜✐Ĩ✉ ❞✐Ơ♥ [2n + 1] q✉❛ ❝➳❝ ✹✻ ❈❤➢➡♥❣ ✷✳ ❇✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ✈➭ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ❝ã ❝❤✐Ị✉ t❤✃♣ ✷ ✹✼ ❈➳❝ ❝➠♥❣ t❤ø❝ t❤ø❝ tÝ♥❤ t♦➳♥ ➤➢ỵ❝ sù ❞ơ♥❣ ë tr➟♥ ➤➢ỵ❝ s✉② r❛ tõ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ tÝ❝❤ t❡♥s♦r✳ ➜è✐ ✈í✐ ❝➳❝ ➤➵✐ sè ▲✐❡ sln ✈í✐ n > t❛ ❜á q✉❛ ♣❤Ð♣ ❧✃② ➤è✐ ♥❣➱✉ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥✱ ❤➡♥ ♥÷❛ t❛ ❝ị♥❣ ❜á q✉❛ ♣❤Ð♣ ❧✃② tÝ❝❤ ♣❤➞♥ ➤è✐ ①ø♥❣✳ ➜Ĩ ♣❤➞♥ ❜✐Ưt ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❦❤➳❝ ♥❤❛✉✱ t❛ sÏ ♣❤➞♥ tÝ❝❤ ❝❤ó♥❣ q✉❛ ❧ò② t❤õ❛ ❜❐❝ ❤❛✐ ➤è✐ ①ø♥❣ ✭sym2 ✮ ✈➭ ❧ò② t❤õ❛ ❜❐❝ ❤❛✐ ❝ñ❛ tÝ❝❤ ♥❣♦➭✐ ✭Λ2 ✮✳ ❚❛ tÝ♥❤ t♦➳♥ ➤è✐ ✈í✐ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ♠➭ ❝❤ó♥❣ ❝ã t❤Ĩ ➤➢ỵ❝ ❜✐Ĩ✉ ❞✐Ơ♥ q✉❛ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❝ã ❝❤✐Ị✉ t❤✃♣ ❤➡♥✳ ❑Õt q✉➯ ➤➢ỵ❝ ❝❤♦ ❜ë✐ ❜➯♥❣ ♣❤➞♥ ❧♦➵✐ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝ã ❝❤✐Ị✉ d ≤ ♥❤➢ s❛✉✿ d ➜➵✐ sè ▲✐❡ ❇✐Ĩ✉ ❞✐Ơ♥ Λ2 sym2 ✷ ✸ ✸ ✹ ✹ ✹ ✹ ✺ ✺ ✺ ✻ ✻ ✻ ✻ ✻ ✻ ✻ sl2 sl2 sl3 sl2 sl4 sl4 sl2 × sl2 sl2 sp4 sl5 sl2 sl3 sl4 sl6 sp6 sl2 × sl2 sl2 × sl3 ❬✶❪ ❬✷❪ ❬✶✱✵❪ ❬✸❪ ❬✶✱✵✱✵❪ ❬✶✱✵❪ ❬✵❪ ❬✷❪ ❬✵✱✶❪ ❬✹❪✱ ❬✵❪ ❬✵✱✶✱✵❪ ❬✵✱✶❪✱ ❬✵✱✵❪ ❬✷❪ ❬✹❪✱ ❬✵❪ ❬✷✱✵❪ ❬✻❪✳ ❬✷❪ ❬✷✱✵✱✵❪ ❬✷✱✵❪ [1] ⊗ [1] [0] ⊗ [2], [2] ⊗ [0] [0] ⊗ [0], [2] ⊗ [2] ❬✹❪ ❬✵✱ ✶❪ ❬✶✱ ✵✱ ✵✱ ✵❪ ❬✺❪ ❬✷✱ ✵❪ ❬✵✱✶✱✵❪ ❬✶✱✵✱✵✱✵✱✵❪ ❬✶✱✵✱✵❪ ❬✻❪✱ ❬✷❪ ❬✷✱ ✵❪ ❬✵✱ ✶✱ ✵ ✱✵❪ ❬✽❪✱ ❬✹❪✱ ❬✵❪ ❬✷✱ ✶❪ ❬✶✱✵✱✶❪ ❬✵✱✶✱✵✱✵✱✵❪ ❬✵✱✶✱✵❪✱ ❬✵✱✵✱✵❪ ❬✽❪✱❬✹❪✱ ❬✵❪ ❬✵✱ ✷❪✱ ❬✵✱ ✵❪ ❬✷✱ ✵✱ ✵✱ ✵❪ ❬✶✵❪✱ ❬✻❪✱ ❬✷❪ ❬✹✱ ✵❪✱ ❬✵✱ ✷❪ ❬✵✱✷✱✵❪✱ ❬✵✱✵✱✵❪ ❬✷✱✵✱✵✱✵✱✵❪ ❬✷✱✵✱✵❪ [1] ⊗ [2] [0] ⊗ [0], [0] ⊗ [4], [2] ⊗ [2] [0] ⊗ [2], [2] ⊗ [0], [2] ⊗ [4] [1] ⊗ [1, 0] [0] ⊗ [2, 0], [2] ⊗ [0, 1] [0] ⊗ [0, 1], [2] ⊗ [2, 0] ❙❛✉ ➤➞② ❝❤♦ t❛ ❜➯♥❣ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝ã ❝❤✐Ị✉ tõ • sl3 ❜✐Ĩ✉ ❞✐Ơ♥ [1, 1] ✭❞✐♠ 8), • sl4 ❜✐Ĩ✉ ❞✐Ơ♥ [2, 0, 0](dim 10) • sl5 ❜✐Ĩ✉ ❞✐Ơ♥ [0, 1, 0, 0] ✭❞✐♠ 10) ❜✐Ĩ✉ ❞✐Ơ♥ • so7 ❜✐Ĩ✉ ❞✐Ơ♥ [0, 0, 1] ✭❞✐♠ 8) • sp7 ❜✐Ĩ✉ ❞✐Ơ♥ [2, 0] ✭❞✐♠ 10) [3, 0] ✭❞✐♠ 10) − 11✳ ❈❤➢➡♥❣ ✷✳ ❇✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ✈➭ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ ó ề t ã sl2 ì sl2 ể ễ [1] ⊗ [3] ✭❞✐♠ 8), ✈í✐ ❜✐Ĩ✉ ❞✐Ơ♥ ❜✐Ĩ✉ ❞✐Ơ♥ [1] [4] 10) ã sl2 ì sl3 ể ễ [2] [1, 0] 9) ã sl2 ì sl4 ❜✐Ĩ✉ ❞✐Ơ♥ [1] ⊗ [1, 0, 0] ✭❞✐♠ 8) • sl2 × sp4 ❜✐Ĩ✉ ❞✐Ơ♥ [1] ⊗ [1, 0] 8), ã sl2 ì sl5 ể ễ ã sl3 × sl3 ❜✐Ĩ✉ ❞✐Ơ♥[1, 0] ⊗ [1, 0] ✭❞✐♠ • sl2 × sl2 × sl2 [2] ⊗ [2] ✭❞✐♠ 9), ❜✐Ĩ✉ ❞✐Ơ♥ [1] ⊗ [1, 0, 0, 0] ✭❞✐♠ 10) ❜✐Ĩ✉ ❞✐Ơ♥ 9) [1] ⊗ [1] ⊗ [1] ✭❞✐♠ 8) [1] ⊗ [0, 1] ✭❞✐♠ 10) ✹✽ ❑Õt ❧✉❐♥ ▲✉❐♥ ✈➝♥ ➤➲ t×♠ ❤✐Ĩ✉ ✈➭ tr×♥❤ ❜➭② ♠ét sè ✈✃♥ ➤Ị s❛✉✿ sl2 ✶✳ ❚r×♥❤ ❜➭② tỉ♥❣ q✉❛♥ ✈Ị ❧ý t❤✉②Õt ➤➵✐ sè ▲✐❡ ✈➭ ❝➳❝ ❦❤➳✐ ♥✐Ö♠ ❧✐➟♥ q✉❛♥✳ ✷✳ ❚r×♥❤ ❜➭② ❝❤✐ t✐Õt ❜✐Ĩ✉ ❞✐Ơ♥ ❝đ❛ sl2 , ♠➠ t➯ ❝❤✐ t✐Õt ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝đ❛ t❤❡♦ trä♥❣ ✈➭ ❤Ư ♥❣❤✐Ư♠✳ ✸✳ ❚r×♥❤ ❜➭② ❝❤✐ t✐Õt ❧ý t❤✉②Õt ❝✃✉ tró❝ ❝đ❛ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥✳ ✹✳ ❚r×♥❤ ❜➭② ❝❤✐ t✐Õt ❧ý t❤✉②Õt ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝đ❛ ❝➳❝ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ t❤❡♦ trä♥❣ ✈➭ ✈❡❝t➡ trä♥❣✱ ♣❤➞♥ ❧♦➵✐ ❝➳❝ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥✳ ✺✳ ❙ư ❞ơ♥❣ ♣❤➬♥ ♠Ị♠ trù❝ t✉②Õ♥ ▲✐❡ ➤Ó tÝ♥❤ t♦➳♥ ❧➵✐ ✈➭ ❧❐♣ ❧➵✐ ❜➯♥❣ ❦Õt q✉➯ ✈Ị ❧✐Ưt ❦➟ ❜✐Ĩ✉ ❞✐Ơ♥ ❜✃t ❦❤➯ q✉② ❝ã sè ❝❤✐Ò✉ ❦❤➠♥❣ q✉➳ ❝ã sè ❝❤✐Ò✉ t❤✃♣✳ ✹✾ 11 ❝❤♦ ♠ét ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ❚➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ế ệt ỗ ọ ệ ý tết ó ▲✐❡✱ ❇➭✐ ❣✐➯♥❣ ❙❛✉ ➤➵✐ ❤ä❝✱ ❱✐Ö♥ t♦➳♥ ❤ä❝ ❱✐Öt ◆❛♠✳ ❬✷❪ ◆❣✉②Ô♥ ❚❤Õ ❍♦➭♥✱ P❤➵♠ P❤✉ ✭✷✵✵✸✮✱ ❈➡ së trì ý tết ổ ị ①✉✃t ❜➯♥ ❣✐➳♦ ❞ô❝✳ ❚✐Õ♥❣ ❆♥❤ ❬✸❪✳ ❆✳ ▼✳ ❈♦❤❡♥ ❡t ❛❧✳ ▲✐❊ ❖♥❧✐♥❡ ❙❡r✈✐❝❡✳ ❲❡❜s✐t❡ ❤tt♣✿✴✴✇✇✇✲♠❛t❤✳✉♥✐✈✲♣♦✐t✐❡rs✳❢r✴ ♠❛❛✈❧✴▲✐❊✴✳ ❬✹❪✳ ❇✳ ❈✳ ❍❛❧❧ ✭✷✵✵✸✮✱ ▲✐❡ ❆❧❣❡❜r❛s ❛♥❞ ❘❡♣r❡s❡♥t❛t✐♦♥✱ ❙♣r✐♥❣❡r✳ ❬✺❪✳ ❑✳ ❆✳ ◆❣✉②❡♥ ❛♥❞ ▼✳ ✈❛♥ ❞❡ P✉t ✭✷✵✶✵✮✱ ✧ ❙❧♦✈✐♥❣ ▲✐♥❡❛r ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛✲ t✐♦♥s✧✱ P✉r❡ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ◗✉❛rt❡r❧②✱ ❱♦❧✉♠❡ ✻✱ ◆✉♠❜❡r ✶✱ ♣♣✳ ✶✼✸ ✲ ✷✵✽✳ ❬✻❪✳ ■✳ ●r♦❥♥♦✇s❦✐ ✭✷✵✶✵✮✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ▲✐❡ ❛❧❣❡❜r❛s ❛♥❞ ❚❤❡✐r ❘❡♣r❡s❡♥t❛t✐♦♥✱ ❖♥❧✐♥❡ ▲❡❝t✉r❡ ◆♦t❡s✱ ❯♥✐✈❡rs✐t② ♦❢ ❈❛♠❜r✐❞❣❡✳ ❬✼❪✳ ❱✳ ❑❛❝ ✭✶✾✽✺✮✱ ■♥❢✐♥✐t❡ ✲ ❞✐♠❡♥s✐♦♥❛❧ ▲✐❡ ❛❧❣❡❜r❛s✱ ❈❛♠❜r✐❣❞❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ◆❡✇ ❨♦r❦✳ ✺✵ ... ❝➳❝❤ t❤ø❝ ♥❤➢ ❝➳❝ ♥❤ã♠ ị ợ sử ụ tr ý tết s ể ♣❤➞♥ tÝ❝❤ ❝➳❝ ➤è✐ ①ø♥❣ rê✐ r➵❝ ❝ñ❛ ❝➳❝ ♣❤➢➡♥❣ trì số ỗ ể tứ t ế t ➤é♥❣ ❝ñ❛ ♥❤ã♠ ▲✐❡ ❝❤♦ ♠ét tÝ❝❤ ♣❤➞♥ ➤➬✉ ❝ñ❛ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥✱ ❞♦ ✈❐② ❝❤♦... tư t✉②Õ♥ tÝ♥❤ tr➟♥ C ❳Ðt A = End(V ) ➤➵✐ sè C− ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ V ❑❤✐ ➤ã End(V ) trë t❤➭♥❤ ♠ét số tí ợ ị [f, g] = g.f − f.g, ∀f, g ∈ End(V ), tr♦♥❣ ➤ã ♣❤Ð♣ ♥❤➞♥ ✬✬✳✬✬ ❧➭ ♣❤Ð♣ ❤ỵ♣ t❤➭♥❤ ❤❛✐... t❤➭♥❤ ♠ét ♣❤➵♠ trï ✈í✐ ❝➳❝ ❝✃✉ ①➵ ❝❤Ý♥❤ ❧➭ ❝➳❝ ➤å♥❣ ❝✃✉ ➤➵✐ sè ▲✐❡✳ ◆Õ✉ φ : G1 −→ G2 , ❧➭ ➤å♥❣ ❝✃✉ số tì Ker ủ G1 ò Imφ G2 ❚❤❐t ✈❐②✿ ❧➭ ➤➵✐ sè ❝♦♥ ❝ñ❛ ∀x ∈ Kerφ, y ∈ G1 , t❛ ❝ã [y, x] ∈ Kerφ,