❇é ●✐➳♦ ❉ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❚❤Þ ❚❤➯♦ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ị✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ ▲✉❐♥ ✈➝♥ t❤➵❝ sÜ t♦➳♥ ❤ä❝ ◆❣❤Ö ❆♥ ✲ ✷✵✶✼ ❇é ●✐➳♦ ❉ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❚❤Þ ❚❤➯♦ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ị✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ ▲✉❐♥ ✈➝♥ t❤➵❝ sÜ t♦➳♥ ❤ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ➜➵✐ sè ✈➭ ▲ý t❤✉②Õt sè ▼➲ sè✿ ✻✷ ✹✻ ✵✶ ✵✹ ◆❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ ❚❙✳ ◆❣✉②Ơ♥ ◗✉è❝ ❚❤➡ ◆❣❤Ư ❆♥ ✲ ✷✵✶✼ ▼ơ❝ ❧ơ❝ ▲ê✐ ♥ã✐ ➤➬✉ ✸ ✶ ✽ ✷ ➜➵✐ sè ▲✐❡ ✶✳✶ ➜➵✐ sè ▲✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷ ➜➵✐ sè ▲✐❡ ❧ò② ❧✐♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✸ ➜➵✐ sè ▲✐❡ ❣✐➯✐ ➤➢ỵ❝✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✹ ➜➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ ✸✶ ✷✳✶ ➜➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✷ ➜➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤Þ❛ ♣❤➢➡♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✸ ▼ét sè ❧í♣ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷✳ ✳ ✳ ✳ ✹✷ ❑Õt ❧✉❐♥ ✺✵ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✺✶ ✷ ▲ê✐ ♥ã✐ ➤➬✉ ✶✳ ▲ý ❞♦ ❝❤ä♥ ➤Ò t➭✐ ▲ý t❤✉②Õt ◆❤ã♠ ▲✐❡ ➤➢ỵ❝ ➤➷t t➟♥ t❤❡♦ ♥❤➭ t♦➳♥ ❤ä❝ ♥❣➢ê✐ ◆❛ ❯② ❧➭ ❙♦♣❤✉s ▲✐❡ ✭✶✽✹✷ ✲ ✶✽✾✾✮✱ ❧➭ ❦❤➳✐ ♥✐Ư♠ tỉ♥❣ ❤ß❛ tõ ❤❛✐ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ❧➭ ♥❤ã♠ ✭tr♦♥❣ ➜➵✐ sè✮ ✈➭ ➤❛ t➵♣ ✈✐ ♣❤➞♥ ✭tr♦♥❣ ❍×♥❤ ❤ä❝ ✲ ❚➠♣➠✮✳ ◆❤ã♠ ▲✐❡ ❦❤➠♥❣ ❝❤Ø ❧➭ ❝➠♥❣ ❝ơ ❝đ❛ ❣➬♥ ♥❤➢ t✃t ❝➯ ❝➳❝ ♥❣➭♥❤ ❚♦➳♥ ❤✐Ư♥ ➤➵✐✱ ♠➭ ♥ã ❧➭ ❝➠♥❣ ❝ơ ➤Ĩ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ♥❣➭♥❤ ❝đ❛ ❱❐t ❧ý ❧ý t❤✉②Õt ❤✐Ư♥ ➤➵✐✱ ➤➷❝ ❜✐Ưt ❧➭ ❧ý t❤✉②Õt ❝➳❝ ❤➵t✳ ▼ét tr♦♥❣ ♥❤÷♥❣ ý t➢ë♥❣ ❝đ❛ ❧ý t❤✉②Õt ♥❤ã♠ ▲✐❡ ❧➭ t❤❛② t❤Õ ❝✃✉ tró❝ ♥❤ã♠ t♦➭♥ ❝ơ❝ ❜ë✐ ♣❤✐➟♥ ❜➯♥ ♠❛♥❣ tÝ♥❤ ➤Þ❛ ♣❤➢➡♥❣ ủ ó ò ọ ợ ❧➭♠ t✉②Õ♥ tÝ♥❤ ❤ã❛✳ ❙♦♣❤✉s ▲✐❡ ❣ä✐ ➤ã ❧➭ ♥❤ã♠ ▲✐❡ ✈➠ ❝ï♥❣ ❜Ð✳ ❙❛✉ ♥➭② ♥❣➢ê✐ t❛ ❣ä✐ ➤ã ❧➭ ➜➵✐ sè ▲✐❡✳ ◆❣❤✐➟♥ ❝ø✉ ✈Ò ❝➳❝ ➤➵✐ sè ▲✐❡ tỉ♥❣ q✉➳t ♥ã✐ ❝❤✉♥❣ ✈➭ ❧í♣ ❝➳❝ ➤➵✐ sè ▲✐❡ ❝ơ t❤Ĩ ♥ã✐ r✐➟♥❣ ❧➭ ♠ét ❧Ü♥❤ ✈ù❝ ♥❣❤✐➟♥ ❝ø✉ ré♥❣ tr♦♥❣ ❚♦➳♥ ❤ä❝ ✈➭ ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ tr♦♥❣ ❱❐t ❧ý✳ ❈ơ t❤Ĩ ♥❤➢ ➤➵✐ sè ▲✐❡ ♥ư❛ ột tr ữ ụ ữ ệ ợ sử ❞ơ♥❣ ❦❤➳ ♥❤✐Ị✉ tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ❧➭ ❞➵♥❣ ❑✐❧❧✐♥❣ ♥❤ê ✈➭♦ ❜❛ tÝ♥❤ ❝❤✃t✿ ➤è✐ ①ø♥❣✱ ❜✃t ❜✐Õ♥ ✈➭ ❦❤➠♥❣ s✉② ❜✐Õ♥ ❝ñ❛ ♥ã✳ ❈❤➻♥❣ ❤➵♥ t✐➟✉ ❝❤✉➮♥ ❈❛rt❛♥ tr♦♥❣ ❜➭✐ t♦➳♥ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ ♥ã✐ r➺♥❣ ❑✐❧❧✐♥❣ ❦❤➠♥❣ s✉② ❜✐Õ♥ tr➟♥ ❦❤➠♥❣ ữ số s ế tế tì G G G ❧➭ ➤➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❞➵♥❣ G × G ❉♦ ➤ã ♥❣➢ê✐ t❛ ➤➷t r❛ ♠ét ❝➞✉ ❤á✐✿ ❚å♥ t➵✐ ❤❛② ♠➭ tr➟♥ ➤ã ❝ã ♠ét ❞➵♥❣ s♦♥❣ t✉②Õ♥ tÝ♥❤ ➤è✐ ①ø♥❣✱ ❜✃t ❜✐Õ♥ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ➤➵✐ sè ▲✐❡ G ❝ã ♠ét ❞➵♥❣ s♦♥❣ t✉②Õ♥ tÝ♥❤ ♥❤➢ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣✳ ◆❤÷♥❣ ❝➞✉ ❤á✐ ①♦❛② q✉❛♥❤ ❝➳❝ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤➲ ➤➢ỵ❝ ➤➷t r❛ tõ ❧➞✉✱ ♥❤➢♥❣ ❣➬♥ ➤➞② ♠í✐ ➤➢ỵ❝ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉ ❦❤✐ ①✉✃t ❤✐Ư♥ ♥❤✐Ị✉ ❝➠♥❣ ❝ơ ❞➭♥❤ ❝❤♦ ❝❤ó♥❣ ✭①❡♠ ❬✹❪✱ ❬✻❪✱ ❬✼❪✮ ❝ò♥❣ ♥❤➢ ♥❣➢ê✐ t❛ t❤✃② ♠è✐ ❧✐➟♥ ❤Ư ❝đ❛ ❝❤ó♥❣ tr♦♥❣ ♠ét sè ❜➭✐ t♦➳♥ ❱❐t ❧ý ✭①❡♠ ❬✺❪✮✳ ➜➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤➢ỵ❝ ➳♣ ❞ơ♥❣ tr♦♥❣ ♥❤✐Ò✉ ❧Ü♥❤ ✸ ✹ ▲ê✐ ♥ã✐ ➤➬✉ ✈ù❝ ❚♦➳♥ ❤ä❝ ✈➭ ❱❐t ❧ý✱ ❤✐Ĩ✉ ✈Ị ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❣✐ó♣ ❝❤ó♥❣ t❛ ❤✐Ĩ✉ râ ❤➡♥ ✈Ị ❝✃✉ tró❝ P♦✐ss♦♥ trù❝ ❣✐❛♦✱ ♥❤ã♠ ▲✐❡ P♦✐ss♦♥✱✱✱✳ ❚r➟♥ ❝➡ së ➤➵✐ sè ▲✐❡ ✈í✐ ♠ét ❜✃t ❜✐Õ♥ ➤➢ỵ❝ ❜ỉ s✉♥❣✱ t❛ ①➞② ❞ù♥❣ ➤➢ỵ❝ ♥❤✐Ị✉ ❧í♣ ❝➳❝ ❝✃✉ tró❝ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❝ơ t❤Ĩ ♥❤➢✿ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ◆♦✈✐❦♦✈✱ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❣✐➯✐ ➤➢ỵ❝✱ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤è✐ ♥❣➱✉✱✳✳✳✳✳ ▼ét tr♦♥❣ ♥❤÷♥❣ ❝➠♥❣ ❝ơ ➤➢ỵ❝ sư ❞ơ♥❣ ♥❤✐Ị✉ ➤Ĩ ♥❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❧➭ ♣❤➢➡♥❣ ♣❤➳♣ ♠ë ré♥❣ ❦Ð♣ ➤➢ỵ❝ ➤➢❛ r❛ ➤➬✉ t✐➟♥ ✈➭♦ tr♦♥❣ ❝✉è♥ s➳❝❤ ❝❤✉②➟♥ ❦❤➯♦ ✭①❡♠ ❬✻❪✮ ❝ñ❛ ❱✳ ❑❛❝✳ ➜➞② ❧➭ ♠ét ết ợ ữ rộ t tí trù❝ t✐Õ♣ ❝đ❛ ❝➳❝ ➤➵✐ sè ▲✐❡✳ ❱Ị ♠➷t ❤×♥❤ ➯♥❤✱ ♥Õ✉ ❝❤♦ tr➢í❝ ♠ét ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ G t❛ sÏ ❣➽♥ t❤➟♠ ❤❛✐ ➤➬✉ ❝ñ❛ G ❜ë✐ ♠ét ➤➵✐ sè ▲✐❡ H ✈➭ ❦❤➠♥❣ ❣✐❛♥ ➤è✐ ♥❣➱✉ H∗ H ➤Ĩ ➤➢ỵ❝ ♠ét ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ♠í✐✳ ❈❤✃t ❦❡♦ ➤Ĩ ❣➽♥ ❦Õt ❝➳❝ ❦❤➠♥❣ ❝đ❛ ❣✐❛♥ ♥➭② ❝❤Ý♥❤ ❧➭ ♠ë ré♥❣ t➞♠ ✈➭ tÝ❝❤ ♥ö❛ trù❝ t✐Õ♣✳ ◆➝♠ ✶✾✽✺ ▼❡❞✐♥❛ ✈➭ ❘❡✈♦② ❞ù❛ ✈➭♦ ❦❤➳✐ ♥✐Ö♠ ♠ë ré♥❣ ❦Ð♣ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ ♠ä✐ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤Ị✉ ❧➭ ♠ë ré♥❣ ❦Ð♣ ❝đ❛ ♠ét ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❜ë✐ ♠ét ➤➵✐ sè ▲✐❡ ➤➡♥ ❤♦➷❝ ❜ë✐ ♠ét ➤➵✐ sè ▲✐❡ 1− ❝❤✐Ò✉✳ ◆❣♦➭✐ r❛✱ ❝ị♥❣ ❞ù❛ ✈➭♦ ❦❤➳✐ ♥✐Ư♠ ♠ë ré♥❣ ❦Ð♣ ♥❣➢ê✐ t ò ứ ợ ột số t ♣❤➢➡♥❣ ❣✐➯✐ ➤➢ỵ❝ ➤➢ỵ❝ tõ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ t✐Õ♣ ✈í✐ ♠ét ➤➵✐ sè (n − 2)− ❝❤✐Ị✉ ❜ë✐ ➤➵✐ sè n− 1− ❝❤✐Ị✉ ❝ã t❤Ĩ ♥❤❐♥ ❝❤✐Ị✉ tÝ❝❤ ♥ư❛ trù❝ 1− ❝❤✐Ị✉ ❦❤➳❝✳ ❉♦ ➤ã ♥❤✐Ị✉ ♥❣➢ê✐ ①❡♠ ♠ë ré♥❣ ❦Ð♣ ♥❤➢ ❧➭ ♠ét ❦✐Ó✉ ♠➠ t➯ q✉② ♥➵♣ ❤♦➷❝ ♠ét ❦✐Ĩ✉ ♠ë ré♥❣ ♥❤✐Ị✉ ❜➢í❝ ❝➳❝ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣✳ ❑❤➳✐ ♥✐Ö♠ ♠ë ré♥❣ ❦Ð♣ ➤ã♥❣ ột trò q trọ ì ó sở ❝❤♦ ♣❤➢➡♥❣ ♣❤➳♣ ♣❤➞♥ ❧♦➵✐ q✉② ♥➵♣ ❝➳❝ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣✳ ▼ét ♣❤➢➡♥❣ ♣❤➳♣ ❦❤➳❝ ❝ò♥❣ tá r❛ ❤✐Ö✉ q✉➯ tr♦♥❣ ✈✐Ö❝ ♥❣❤✐➟♥ ❝ø✉ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❣✐➯✐ ➤➢ỵ❝ ❧➭ ♠ë ré♥❣ T∗ ➤➢ỵ❝ ❣✐í✐ t❤✐Ư✉ ❜ë✐ ▼✳ ❇♦r❞❡♠❛♥♥ ✈➭♦ ♥➝♠ ✶✾✾✼ T∗ ❧➭ ❦✐Ó✉ ♠ë ré♥❣ ♠ét ❜➢í❝ ❝đ❛ ❝➳❝ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣✳ ❚õ ♠ét ➤➵✐ sè ▲✐❡ ❜✃t ❦ú G, t❛ ❣➽♥ t❤➟♠ ❦❤➠♥❣ ✭①❡♠ ❬✹❪✮✳ ❑❤➳❝ ✈í✐ ♣❤➢➡♥❣ ♣❤➳♣ ♠ë ré♥❣ ❦Ð♣✱ ♠ë ré♥❣ ❣✐❛♥ ➤è✐ ♥❣➱✉ ❝ñ❛ ♥ã ♥❤ê tÝ❝❤ ♥ư❛ trù❝ t✐Õ♣ ❜ë✐ ❜✐Ĩ✉ ❞✐Ơ♥ ➤è✐ ♣❤ơ ❤ỵ♣ ❝đ❛ G∗ ✈➭ ♠ét G tr♦♥❣ 2− ➤è✐ ❝❤✉ tr×♥❤ ❝②❝❧✐❝✳ ❑❤✐ ➤ã t❛ t❤✉ ➤➢ỵ❝ ♠ét ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣✳ ◆❤➢ t❤Õ ❝ã t❤Ó ①❡♠ ♠ë ré♥❣ T ∗ ♥❤➢ ❧➭ ♠ét ❦✐Ĩ✉ ❦❤➳✐ q✉➳t ❝đ❛ tÝ❝❤ ♥ư❛ trù❝ t✐Õ♣ ❝đ❛ ♠ét ➤➵✐ sè ▲✐❡ ✈í✐ ♠ét ❦❤➠♥❣ ❣✐❛♥ ➤è✐ ♥❣➱✉ ❝đ❛ ♥ã ❜ë✐ ❜✐Ĩ✉ ❞✐Ơ♥ ➤è✐ ♣❤ơ ❤ỵ♣✳ ❉ù❛ ✺ ▲ê✐ ♥ã✐ ➤➬✉ tr➟♥ ❦❤➳✐ ♥✐Ö♠ ♥➭②✱ ➠♥❣ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ ♠ä✐ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❣✐➯✐ ➤➢ỵ❝ tr➟♥ tr➢ê♥❣ ➤ã♥❣ ➤➵✐ sè ❝ã ➤➷❝ sè ❜➺♥❣ s✉② ❜✐Õ♥ ❝ã sè ➤è✐ ❝❤✐Ò✉ ❜➺♥❣ ❧➭ ♠ë ré♥❣ T ∗ ❤♦➷❝ ❧➭ ✐❞❡❛❧ ❦❤➠♥❣ ❚r♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ❝❤ó♥❣ t➠✐ t✐Õ♣ ❝❐♥ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ t❤❡♦ ❤➢í♥❣ q✉❡♥ t❤✉é❝✱ ➤ã ❧➭ ♥❣❤✐➟♥ ❝ø✉ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❝ã sè ❝❤✐Ò✉ ❜Ð✳ tế ó ợ ỗ ó tể ①❡♠ ①Ðt ♥❤✐Ị✉ ❦❤➳✐ ♥✐Ư♠ ♣❤ø❝ t➵♣ ❝đ❛ ❧í♣ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ tr➟♥ ♥❤÷♥❣ ✈Ý ❞ơ ❝ơ t❤Ĩ ❝ã sè ❝❤✐Ị✉ ❜Ð ✈➭ s❛✉ ➤ã tỉ♥❣ q✉➳t trë ệ ó ột t ợ ữ t❤➠♥❣ q✉❛ ✈✐Ö❝ ♣❤➞♥ ❧♦➵✐ ❤❛② ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ tÝ♥❤ ❝❤✃t ➤➳♥❣ ❝❤ó ý tr➟♥ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❝ã sè ❝❤✐Ị✉ ❜Ð✱ ❝❤ó♥❣ t❛ ❤② ✈ä♥❣ sÏ ♣❤➳t ❤✐Ư♥ ♥❤✐Ị✉ ❧í♣ ❝♦♥ ➤➷❝ ❜✐Ưt ❝đ❛ ❧í♣ ❝➳❝ ➤➵✐ số t ũ tì t ữ ❝ơ ♥❣❤✐➟♥ ❝ø✉ ♠í✐✳ ❈ơ t❤Ĩ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝➳❝ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❝ơ t❤Ĩ ❝ã sè ❝❤✐Ị✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ❱✐Ö❝ ♥❣❤✐➟♥ ❝ø✉ ✈➭ ♣❤➞♥ ❧♦➵✐ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ♥ã✐ ❝❤✉♥❣ ✈➭ ❧í♣ ❝➳❝ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❝ã sè ❝❤✐Ò✉ ❜Ð ♥ã✐ r✐➟♥❣ ➤❛♥❣ ➤➢ỵ❝ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ tr♦♥❣ ✈➭ ♥❣♦➭✐ ♥➢í❝ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉✳ ❱í✐ ♠♦♥❣ ♠✉è♥ t×♠ ❤✐Ĩ✉ ✈Ị ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣✱ ❝❤ó♥❣ t➠✐ ❝❤ä♥ ➤Ị t➭✐✿ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ ❧➭♠ ➤Ị t➭✐ ❧✉❐♥ ✈➝♥✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ❧➭ ❞ù❛ ✈➭♦ t➭✐ ❧✐Ö✉ ❬✸❪✱ ➤ã ❧➭ ♥é✐ ❞✉♥❣ ❝ñ❛ ❜➭✐ ❜➳♦ ✧▲✐❡ ❆❧❣❡❜r❛s ✇✐t❤ q✉❛❞r❛t✐❝ ❞✐♠❡♥s✐♦♥ ❡q✉❛❧ t♦ ✷✧✱ ❏♦✉r♥❛❧ ♦❢ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ❆❧✲ ❣❡❜❛r❛✱ ✷✵✾✭✷✵✵✼✮✱ ♣♣✳ ✼✷✺ ✲ ✼✸✼ ❝ñ❛ ❝➳❝ t➳❝ ❣✐➯ ■❣♥❛❝✐♦ ❇❛❥♦ ✈➭ ❙❛✐❞ ❇❡♥❛②❛❞✐ ✈➭ ❝➳❝ t➭✐ ❧✐Ư✉ ❧✐➟♥ q✉❛♥ ➤Ĩ ➤ä❝ ❤✐Ĩ✉✱ tr×♥❤ ❜➭② ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❝ã sè ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷✳ ◆é✐ ❞✉♥❣ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ ❧✉❐♥ ✈➝♥ ✷✳✶✳ ❚r×♥❤ ❜➭② ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈Ị ➤➵✐ sè ▲✐❡✱ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❝ã sè ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✈➭ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧✐➟♥ q✉❛♥✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ♠ét sè ❦Õt q✉➯ ✈Ò ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❝ã sè ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷✳✷✳ ❜➺♥❣ ❚❤Ĩ ❤✐Ư♥ t➢ê♥❣ ♠✐♥❤ ❝➳❝ ❦Õt q✉➯ tr➟♥ ❝❤♦ ❧í♣ ➤➵✐ sè ▲✐❡ ❝ã ❝❤✐Ị✉ t♦➭♥ ♣❤➢➡♥❣ 2✱ ❝ơ t❤Ĩ ♥❤➢✿ ➜➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤Þ❛ ♣❤➢➡♥❣✱ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❧ò② ❧✐♥❤✱ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤➬② ➤đ✱✳✳✳ ✻ ▲ê✐ ♥ã✐ ➤➬✉ ✸✳ ❚ỉ♥❣ q✉❛♥ ✈➭ ❝✃✉ tró❝ ❝đ❛ ❧✉❐♥ ✈➝♥ ◆❣♦➭✐ ♣❤➬♥ ▲ê✐ ♥ã✐ ➤➬✉✱ ❑Õt ❧✉❐♥ ✈➭ ❚➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ t❤× ♥é✐ ❞✉♥❣ ủ ợ trì tr ✶✿ ➜➵✐ sè ▲✐❡✳ ◆é✐ ❞✉♥❣ ❝❤Ý♥❤ tr♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ➤➵✐ sè ▲✐❡✱ ➤➵✐ sè ▲✐❡ ❧ị② ❧✐♥❤✱ ➤➵✐ sè ▲✐❡ ❣✐➯✐ ➤➢ỵ❝✱ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥✳ ❈➳❝ ♥é✐ ❞✉♥❣ ♥ã✐ ë tr➟♥ ➤➢ỵ❝ ❝❤✐❛ t❤➭♥❤ ❝➳❝ t✐Õt s❛✉✿ ✶✳✶✳ ➜➵✐ sè ▲✐❡✳ ✶✳✷✳ ➜➵✐ sè ▲✐❡ ❧ị② ❧✐♥❤✳ ✶✳✸✳ ➜➵✐ sè ▲✐❡ ❣✐➯✐ ➤➢ỵ❝✳ ✶✳✹✳ ➜➵✐ sè ▲✐❡ ♥ö❛ ➤➡♥✳ ❈❤➢➡♥❣ ✷✿ ➜➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷✳ ◆é✐ ❞✉♥❣ ❝❤Ý♥❤ ❝đ❛ ❝❤➢➡♥❣ ♥➭② ❧➭ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ ✈Ò ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣✱ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤Þ❛ ♣❤➢➡♥❣✳ ▼ét sè ❦Õt q✉➯ ❧✐➟♥ q✉❛♥ ➤Õ♥ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❧ò② ❧✐♥❤✱ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤➬② ➤ñ✱ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❣✐➯✐ ➤➢ỵ❝ ❝ã ❝❤✐Ị✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷✳ ❈➳❝ ♥é✐ ❞✉♥❣ ♥ã✐ ë tr➟♥ ➤➢ỵ❝ ❝❤✐❛ t❤➭♥❤ ❝➳❝ t✐Õt s❛✉✿ ✷✳✶✳ ➜➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣✳ ✷✳✷✳ ➜➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤Þ❛ ♣❤➢➡♥❣✳ ✷✳✸✳ ▼ét sè ❧í♣ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷✳ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ ❚❤➬② ❣✐➳♦ ❚❙✳ ◆❣✉②Ơ♥ ◗✉è❝ ❚❤➡✳ ◆❤➞♥ ị t ợ tỏ ò í trä♥❣ ✈➭ ❜✐Õt ➡♥ ❚❤➬②✱ ➤➲ t❐♥ t×♥❤ ❣✐ó♣ ➤ì t➠✐ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ❚➳❝ ❣✐➯ ①✐♥ ➤➢ỵ❝ ❝➯♠ ➡♥ ❝➳❝ ❚❤➬② ✭❈➠✮ tr♦♥❣ ❈❤✉②➟♥ ♥❣➭♥❤ ➜➵✐ sè ✈➭ ▲ý t❤✉②Õt sè ❝đ❛ ❱✐Ư♥ ❙➢ ♣❤➵♠ tù ♥❤✐➟♥✱ ❇❛♥ ●✐➳♠ ❤✐Ư✉ ✈➭ ❝➳❝ P❤ß♥❣ ❜❛♥ ❝❤ø❝ ♥➝♥❣ ❝đ❛ ❚r➢ê♥❣ ➜❍ ❱✐♥❤ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ➤Ĩ t➳❝ ❣✐➯ ❤♦➭♥ t❤➭♥❤ ♥❤✐Ư♠ ✈ơ ❤ä❝ t❐♣✳ ❈➯♠ ➡♥ ❚r➢ê♥❣ ➜❍ ❙➭✐ ●ß♥ ➤➲ tỉ ❝❤ø❝ ✈➭ t➵♦ ➤✐Ị✉ ❦✐Ư♥ ❤ä❝ t❐♣ tèt ❝❤♦ t➳❝ ❣✐➯✳ ❚➳❝ ❣✐➯ ①✐♥ ➤➢ỵ❝ ❝➯♠ ➡♥ ❝➳❝ ❚❤➬② ✭❈➠✮✱ ❝➳❝ ➤å♥❣ ♥❣❤✐Ö♣ ♥➡✐ t➳❝ ❣✐➯ ➤❛♥❣ ❣✐➯♥❣ ❞➵② ✈➭ ❝➠♥❣ t➳❝ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐✱ ❝ỉ ✈ị✱ ➤é♥❣ ✈✐➟♥ ✈➭ ❣✐ó♣ ➤ì t➳❝ ❣✐➯ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❧➭♠ ❧✉❐♥ ✈➝♥ tèt ♥❣❤✐Ö♣✳ ✼ ▲ê✐ ♥ã✐ ➤➬✉ ❈➯♠ ➡♥ sù ❤② s✐♥❤ ❝đ❛ ❣✐❛ ➤×♥❤ ề t ỗ ự t t ữ ể 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✐ ♥❤÷♥❣ t❤✐Õ✉ sãt✳ rt ợ ữ ó ý ❝đ❛ ❝➳❝ ♥❤➭ ❦❤♦❛ ❤ä❝ ✈➭ ➤å♥❣ ♥❣❤✐Ư♣ ➤Ĩ ❧✉❐♥ ✈➝♥ ❝ã t❤Ĩ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥ tèt ❤➡♥✳ ◆❣❤Ư ❆♥✱ ♥❣➭② ✷✵ t❤➳♥❣ ✼ ♥➝♠ ✷✵✶✼ ❚➳❝ ❣✐➯ ◆❣✉②Ơ♥ ❚❤Þ ❚❤➯♦ ❈❤➢➡♥❣ ✶ ➜➵✐ sè ▲✐❡ ◆é✐ ❞✉♥❣ ❝❤Ý♥❤ tr♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❧➵✐ ♠ét sè ❦❤➳✐ ♥✐Ư♠ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈Ị ❞➵♥❣ s♦♥❣ t✉②Õ♥ tÝ♥❤✱ ➤➵✐ sè✱ ➤➵✐ sè ▲✐❡✱ ➤➵✐ sè ▲✐❡ ❧ị② ❧✐♥❤✱ ➤➵✐ sè ▲✐❡ ❣✐➯✐ ➤➢ỵ❝✱ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥✳ ✶✳✶ ➜➵✐ sè ▲✐❡ ✶✳✶✳✶✳ ➜Þ♥❤ ♥❣❤Ü❛✳ t✉②Õ♥ tÝ♥❤ ✐✮ tr➟♥ V ❈❤♦ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ tr➟♥ tr➢ê♥❣ V ❧➭ ➳♥❤ ①➵ K ▼ét ❞➵♥❣ s♦♥❣ B : V × V −→ K, t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿ B(λ1 v1 + λ2 v2 , w) = λ1 B(v1 , w) + λ2 B(v2 , w) ✐✐✮ B(v, β1 w1 + β2 w2 ) = β1 B(v, w1 ) + β2 B(v, w2 ) ✈í✐ ♠ä✐ vi , wi ∈ V ; λi , βi ∈ K; i = 1, ✰✮ ❉➵♥❣ s♦♥❣ t✉②Õ♥ tÝ♥❤ B tr➟♥ V ➤➢ỵ❝ ❣ä✐ ❧➭ ➤è✐ ①ø♥❣ ♥Õ✉ B(v, w) = B(w, v), ∀v, w ∈ V ✰✮ ❉➵♥❣ s♦♥❣ t✉②Õ♥ tÝ♥❤ B tr➟♥ V ➤➢ỵ❝ ❣ä✐ ❧➭ ♣❤➯♥ ➤è✐ ①ø♥❣ ♥Õ✉ B(v, w) = −B(w, v), ∀v, w ∈ V ✰✮ ❈❤♦ ➤ã U⊥ U ❧➭ t❐♣ ❝♦♥ ❝ñ❛ ➜➷t ❧➭ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ ❝♦♥ ❝ñ❛ ❦❤➠♥❣ s✉② ❜✐Õ♥ tr➟♥ ❱ ✰✮ ❈❤♦ ❤➢í♥❣ V B ➤è✐ ✈í✐ ♥Õ✉ U ⊥ = {v ∈ V |B(u, v) = 0, ∀u ∈ U } V ❉➵♥❣ s♦♥❣ t✉②Õ♥ tÝ♥❤ B ♥Õ✉ V ➤➢ỵ❝ ❣ä✐ ❧➭ V ⊥ = {0} ❧➭ ♠ét s♦♥❣ t✉②Õ♥ tÝ♥❤ tr➟♥ B tr➟♥ ❑❤✐ B(v, v) = ✽ V ▼ét ✈❡❝t➡ v ∈ V ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➻♥❣ ❈❤➢➡♥❣ ✶✳ ✾ ➜➵✐ sè ▲✐❡ ❚õ ❝➳❝ ➤Þ♥❤ ♥❣❤Ü❛ tr➟♥ t❛ ❝ã ❝➳❝ ❦Õt q✉➯ s❛✉✿ ✰✮ ●✐➯ sö ❣✐❛♥ ❝♦♥ B ❧➭ ♠ét s♦♥❣ t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ s✉② ❜✐Õ♥ tr➟♥ V ❑❤✐ ➤ã ✈í✐ ♠ä✐ ❦❤➠♥❣ ❝đ❛ U V = U ⊕ U⊥ V, t❛ ❝ã dim(U ) + dim(U ⊥ ) = dim(V ) ◆Õ✉ U ∩ U ⊥ = {0} t❤× ✈➭ t❤✉ ❤Ñ♣ ❞➵♥❣ s♦♥❣ t✉②Õ♥ tÝ♥❤ B tr➟♥ U ✈➭ tr➟♥ U⊥ ❧➭ ❦❤➠♥❣ s✉② ❜✐Õ♥✳ ✰✮ ◆Õ✉ B ❧➭ ♠ét s♦♥❣ t✉②Õ♥ tÝ♥❤ ♣❤➯♥ ①ø♥❣ ✈➭ ➤➷❝ sè ❝đ❛ tr➢ê♥❣ K ❦❤➳❝ ❦❤➠♥❣✳ t❤× ♠ä✐ ✈❡❝t➡ ❝đ❛ ✰✮ ◆Õ✉ B V ❧➭ ♠ét s♦♥❣ t✉②Õ♥ tÝ♥❤ ➤è✐ ①ø♥❣ t❤× ✈❡❝t➡ ❦❤➠♥❣ ❝đ❛ ➤➻♥❣ ❤➢í♥❣ ➤è✐ ✈í✐ ✰✮ ◆Õ✉ t❤× tå♥ t➵✐ B V ❧➭ θV ❧✉➠♥ B ❧➭ ♠ét s♦♥❣ t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ s✉② ❜✐Õ♥ ✈➭ w∈V ✰✮ ❈❤♦ ➤Ị✉ ➤➻♥❣ ❤➢í♥❣✳ s❛♦ ❝❤♦ v∈V ❧➭ ✈❡❝t➡ ➤➻♥❣ ❤➢í♥❣ B(v, w) = ❧➭ ♠ét s♦♥❣ t✉②Õ♥ tÝ♥❤ ➤è✐ ①ø♥❣✱ ❦❤➠♥❣ s✉② ❜✐Õ♥ tr➟♥ V ❑❤✐ ➤ã tå♥ ♥Õ✉ i = j t➵✐ ♠ét ❝➡ së {v1 , v2 , · · · , } ❝ñ❛ V s❛♦ ❝❤♦ B(vi , vj ) = = ♥Õ✉ i = j ✶✳✶✳✷✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ❈❤♦ K ❧➭ ♠ét tr➢ê♥❣ ✈➭ G ❧➭ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ tr➟♥ K ❑❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ B G ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ➤➵✐ sè ▲✐❡ tr➟♥ K ❤❛② K− ➤➵✐ sè ▲✐❡ ♥Õ✉ tr G ợ tr ị ột é ọ tÝ❝❤ ▲✐❡ [., ] : G × G −→ G (x, y) −→ [x, y] s❛♦ ❝❤♦ ❝➳❝ t✐➟♥ ➤Ò s❛✉ ➤➞② t❤á❛ ♠➲♥✿ L1 ❚Ý❝❤ ▲✐❡ ❧➭ t♦➳♥ tö s♦♥❣ t✉②Õ♥ tÝ♥❤✱ tø❝ ❧➭ ∀x, y, z ∈ G, , K, tì [x + ày, z] = λ[x, z] + µ[y, z], [x, λy + µz] = λ[x, y] + µ[x, z], 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 ▲✐❡ G ❝❤Ý♥❤ ❧➭ sè ❝❤✐Ị✉ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ G ❈❤➢➡♥❣ ✷✳ ii) ⇒ i) ✸✼ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ G ❑❤✐ ❧➭ tỉ♥❣ trù❝ t✐Õ♣ ❝đ❛ ❤❛✐ ✐❞❡❛❧ ➤➡♥ ❤♦➷❝ ❧➭ tỉ♥❣ trù❝ t✐Õ♣ ❝đ❛ ♠ét ✐❞❡❛❧ ➤➡♥ ✈➭ ♠ét ✐❞❡❛❧ ♠ét ❝❤✐Ị✉ t❤× ❦❤✐ ➤ã tr➟♥ t❤➢ê♥❣ t❤ù❝ sù ♠➭ ❤➵♥ ❝❤Õ ❝ñ❛ ✐❞❡❛❧ ➤ã ❧➭ ✐❞❡❛❧ ➤➡♥✳ ❱❐② G q✉② ✈➭ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ➤ã ➤➵✐ sè ▲✐❡ ë tr➟♥ ♥ã ✈➱♥ ❝ß♥ ❞➵♥❣ ❦❤➠♥❣ s✉② ❜✐Õ♥✱ ❝ơ t❤Ĩ ❦❤➯ q✉②✳ G (G, B) ❦❤➳❝ ❦❤➠♥❣✱ ❜✃t ❦❤➯ dq (G) = ❧➭ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤Þ❛ ♣❤➢➡♥❣✳ ✭①❡♠ ❬✸❪✮ ❈❤♦ (G, B) ❧➭ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❦❤➠♥❣ ❦❤➯ q✉②✳ ❑❤✐ ❧➭ ➤Þ❛ ♣❤➢➡♥❣ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ 1− ❝❤✐Ò✉ ❤♦➷❝ G ❈❤ø♥❣ ♠✐♥❤✳ ❧✉➠♥ tå♥ t➵✐ ♠ét ✐❞❡❛❧ ❦❤➠♥❣ t➬♠ ✭①❡♠ ❬✸❪✮ ▼ä✐ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ✷✳✷✳✸✳ ➜Þ♥❤ ❧ý✳ ✷✳✷✳✹✳ ➜Þ♥❤ ❧ý✳ B G G ❧➭ ➤➵✐ sè ▲✐❡ ❣✐➯✐ ➤➢ỵ❝ ❝ã t➞♠ ❧➭ ❧➭ ➤➵✐ sè ▲✐❡ ➤➬② ➤đ ✈í✐ ♥❤➞♥ tư ▲✐✈❡ ➤➡♥✳ ❑ý ❤✐Ư✉ Z(G) ❧➭ t➞♠ ❝đ❛ G ➜➷t p(G) ❧➭ sè ❝❤✐Ị✉ ❝đ❛ Z(G)❀ s(G) ❧➭ sè ✐❞❡❛❧ ➤➡♥ ❝đ❛ ❤➵♥❣ tư ▲❡✈✐ ❝đ❛ G ✈➭ m(G) ❧➭ sè ✐❞❡❛❧ ❝ù❝ t✐Ó✉ tr♦♥❣ ♣❤➞♥ tÝ❝❤ ❝đ❛ Soc(G) ●✐➯ sư (G, B) ❧➭ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤Þ❛ ♣❤➢➡♥❣✳ ➜✐Ị✉ ♥➭② ❝ã ♥❣❤Ü❛ ❧➭ Soc(G) ❧➭ ♠ét ✐❞❡❛❧ ❝ù❝ t✐Ó✉ ✈➭ ❞♦ ➤ã s(G) + p(G) = ❚❛ ❝ã ❤❛✐ tr➢ê♥❣ ❤ỵ♣✿ ✰✮ ◆Õ✉ s(G) = ✈➭ p(G) = t❤× G ✰✮ ◆Õ✉ s(G) = ✈➭ p(G) = t❤× G ❧➭ ❣✐➯✐ ➤➢ỵ❝ ✈➭ ❝ã t➞♠ 1− ❝❤✐Ị✉✳ ❧➭ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤➬② ➤đ ✈➭ ❤➵♥❣ tư ▲❡✈✐ ➤➡♥✳ ◆❣➢ỵ❝ ❧➵✐✱ ♥Õ✉ G ❧➭ ➤➵✐ sè ▲✐❡ ❣✐➯✐ ➤➢ỵ❝ ❝ã t➞♠ ➤➬② ➤đ ✈í✐ ❤➵♥❣ tư ▲❡✈✐ ➤➡♥ t❤× ❝❤ó♥❣ t❛ ❝ã ❝ã ♥❣❤Ü❛ ❱í✐ Soc(G) ❧➭ ✐❞❡❛❧ ❝ù❝ t✐Ó✉ ❤❛② G S ❧➭ ➤➵✐ sè ▲✐❡✱ ❦ý ❤✐Ư✉ ❝♦♥ ❜✃t ❜✐Õ♥ ❝đ❛ V ❧➭ ➤➵✐ sè ▲✐❡ m(G) = s(G) + p(G) = ➜✐Ò✉ ♥➭② ❧➭ ➤➵✐ sè ▲✐❡ ➤Þ❛ ♣❤➢➡♥❣✳ V S = {x ∈ V |ψ(s)x = 0, ∀s ∈ S} ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❧✐➟♥ q✉❛♥ ➤Õ♥ ♣❤Ð♣ ❜✐Ĩ✉ ❞✐Ơ♥ ❣✐❛♥ ❝♦♥ t✉②Õ♥ tÝ♥❤ ❝đ❛ 1− ❝❤✐Ị✉ ❤♦➷❝ G ψ : S −→ gl(A), V A ợ ị ➜Þ♥❤ ❧ý s❛✉ ➤➞② ❝❤♦ t❛ ♠ét ❦Õt q✉➯ ✈Ị ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❦❤➠♥❣ ❝❤Ý♥❤ q✉②✳ ✷✳✷✳✺✳ ➜Þ♥❤ ❧ý✳ ✭①❡♠ ❬✸❪✮ ❈❤♦ (G, B) ❧➭ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❦❤➠♥❣ ❦❤➯ q✉②✳ ❑❤✐ ➤ã✿ ✐✮ G ❧➭ ị ợ ỉ t♦➭♥ ♣❤➢➡♥❣ ❧ị② ❧✐♥❤ ❦❤➳❝ ❦❤➠♥❣ t➞♠ Z(A) ❝đ❛ A (G, B) ❧➭ ♠ë ré♥❣ ❦Ð♣ ❝ñ❛ ♠ét ➤➵✐ sè (A, T ) ❜ë✐ t♦➳♥ tö ➤➵♦ ❤➭♠ δ ❦❤➯ ♥❣❤Þ❝❤ tr➟♥ ❈❤➢➡♥❣ ✷✳ G ✐✐✮ ✸✽ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ị✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ (G, B) ❧➭ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ ➤➬② ➤ñ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❧ò② ❧✐♥❤ (A, T ) ❧➭ ♠ë ré♥❣ ❦Ð♣ ❝ñ❛ ♠ét ❜ë✐ ➤➵✐ sè ▲✐❡ ➤➡♥ S q✉❛ ♣❤Ð♣ ❜✐Ĩ✉ ❞✐Ơ♥ ψ : S −→ Dera (A, T ) s❛♦ ❝❤♦ (Z(A))S = {0} ❈❤ø♥❣ ♠✐♥❤✳ ✐✮ ●✐➯ sư G ❝ã ❝❤✐Ị✉ ❜➺♥❣ ✶✳ ❉♦ ➤ã✱ ❱× Z ∈ [G, G], ✈í✐ A = [G, G]/KX Y ∈G ❧➭ ✐❞❡❛❧ ❝ù❝ ➤➵✐ ✈➭ A = [G, G]/KX δ ∈ Dera (A, T ) t♦➳♥ tö ➤➵♦ ❤➭♠ G ❞♦ ➤ã tå♥ t➵✐ [G, G] ❚❤❡♦ ❬✸✱ ➤Þ♥❤ ❧ý ✷❪✱ ♥Õ✉ Z(G) Z(G) = KX, ∀X ∈ G [G, G] = (Z(G))⊥ , ♥➭② ❝❤♦ t❛ t❤✃② r➺♥❣ ọ ị ợ ị ý ✷✳✷✳✹✱ t❛ ❝ã t❤× G s❛♦ ❝❤♦ ➜✐Ị✉ ❧➭ ♠ét ➤➵✐ sè ▲✐❡ ❝♦♥ ❝ñ❛ KY ♣❤➯✐ ❧➭ ♠ë ré♥❣ é ủ ợ ị N : [G, G] −→ A G = [G, G] ⊕ KY G (A, T ) ❜ë✐ δ(N (Z)) = N ([Y, Z]) ✈í✐ ❧➭ ♠ét ♣❤Ð♣ ❜✐Ĩ✉ ❞✐Ơ♥ ❝❤Ý♥❤ t➽❝✳ ❑❤✐ ➤ã✱ ❧➭ ♠ét ➤➵✐ sè ▲✐❡ ❧ò② ❧✐♥❤✱ ❞♦ ✈❐② [G, G] ❝đ❛ ➤➵✐ sè ▲✐❡ ❣✐➯✐ ➤➢ỵ❝ ❝ị♥❣ ❧ị② ❧✐♥❤✳ A = {0} ❚❤❐t ✈❐②✱ ❣✐➯ sư A = {0} t❤× [G, G] = KX ❇➞② ❣✐ê t❛ ❝❤ø♥❣ ♠✐♥❤ ❑❤✐ ➤ã✱ G ❧➭ ➤➵✐ sè ▲✐❡ 2− ❝❤✐Ị✉ ✈➭ ❧ị② ❧✐♥❤✱ ❞♦ ➤ã G ♥➭② ♠➞✉ t❤✉❐♥ ✈í✐ ❣✐➯ t❤✐Õt ❈❤ø♥❣ ♠✐♥❤ ❱í✐ δ G ❧➭ ➤➵✐ sè ▲✐❡ ❣✐❛♦ ❤♦➳♥✳ ➜✐Ị✉ ❧➭ ➤➵✐ sè ▲✐❡ ➤Þ❛ ♣❤➢➡♥❣ ♠➭ ❝ã tí✐ ❦❤➯ ♥❣❤Þ❝❤ tr➟♥ t➞♠ ❝đ❛ δ(N (Z)) = N ([Y, Z]), ∀Z ∈ [G, G] 2− ✐❞❡❛❧ ❝ù❝ t✐Ó✉✳ A ❚❤❐t ✈❐②✿ ✈➭ N : [G, G] −→ A ▲✃② a ∈ [G, G] s❛♦ ❝❤♦ N (a) ∈ Z(A) ✈➭ δ(N (a)) = δ(N (a)) = ✈í✐ [Y, a] ∈ KX = Z(G) ❉♦ ➤ã✱ B([Y, a], [G, G]) = {0}, ✈× [G, G] = (Z(G))⊥ ❇➟♥ ❝➵♥❤ ➤ã✱ ❤❛② ♥➟♥ N ([Y, a]) = ∈ A = [G, G]/KX, ❱× B([Y, a], Y ) = −B(a, [Y, Y ]) = {0} ➤✐Ò✉ ♥➭② t➢➡♥❣ ➤➢➡♥❣ ❙✉② r❛ B([Y, a], G) = {0} [Y, a] = ❱× N (a) ∈ Z(A) ♥➟♥ [a, [G, G]] ⊆ Z(G) ❱❐② B([a, [G, G]], [G, G]) = {0} ▼➷t ❦❤➳❝ ✈× tÝ♥❤ ❝❤✃t ❜✃t ❜✐Õ♥ ❝đ❛ B ♥➟♥ B([a, [G, G]], Y ) = −B([G, G], [a, Y ]) = {0}( ❞♦ [Y, a] = 0) ❈❤➢➡♥❣ ✷✳ B([a, [G, G]], G) = {0}, ➜✐Ò✉ ♥➭② ❞➱♥ ➤Õ♥ a ∈ Z(G) = KX, ❞♦ ✈❐② N (a) = ❈❤ø♥❣ tá δ (G, B) ◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sư ✈í✐ (A, T ) ✸✾ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ A = {0} s✉② r❛ [a, [G, G]] = {0} ❦❤➯ ♥❣❤Þ❝❤ tr➟♥ t➞♠ ❝ñ❛ A ❧➭ ♠ë ré♥❣ ❦Ð♣ ❝ñ❛ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❧ị② ❧✐♥❤ ❜ë✐ t♦➳♥ tư ➤➵♦ ❤➭♠ δ ❦❤➯ ♥❣❤Þ❝❤ tr➟♥ G = Ke∗ ⊕ A ⊕ Ke, ✈í✐ Ke ❧➭ ➤➵✐ sè ▲✐❡ 1− ❝❤✐Ị✉✳ ❱× G Z(A) ❑❤✐ ➤ã✱ t❛ ❝ã ❧➭ tÝ❝❤ ♥ö❛ trù❝ t✐Õ♣ ❝đ❛ ❤❛✐ ➤➵✐ sè ▲✐❡ ❣✐➯✐ ➤➢ỵ❝ ❧➭ A ✈➭ Ke ♥➟♥ G ❝ị♥❣ ❣✐➯✐ ➤➢ỵ❝ ✈➭ ❞♦ ➤ã Soc(G) = Z(G) Z(G) = Ke∗ ❚õ ➤Þ♥❤ ♥❣❤Ü❛ ♠ë ré♥❣ ❦Ð♣ râ r➭♥❣ e∗ ∈ Z(G) ❚❤❐t ❚❛ ❝❤ø♥❣ ♠✐♥❤ ✈❐②✱ ✈í✐ ❤❛② Y ∈ G, t❛ ❝ã [e∗ , Y ] = [e∗ , αe∗ + a + λe] = λ[e∗ , e] = ✈× Ke ❧➭ ➤➵✐ sè 1− ❝❤✐Ị✉✳ ❱❐② Ke∗ ⊆ Z(G) ◆❣➢ỵ❝ ❧➵✐✱ ❧✃② X = αe∗ + a + λe ∈ Z(G) ❱í✐ ▲✐❡ α, λ ∈ K; a ∈ A; b ∈ B, t❛ ❝ã✿ [X, b]G = [αe∗ + a + λe, b] = [a, b]A + T (δ(a), b)e∗ + λδ(b) = ❘â r➭♥❣ t❤× T (δ(a), b) = ❱× λδ(b) = δ ✈➭ [a, b] = −λδ(b) ❦❤➯ ♥❣❤Þ❝❤ tr➟♥ Z(A) ❱❐② ♥Õ✉ t❛ ❝❤ä♥ ♥➟♥ δ(b) = 0 = b ∈ Z(A) s✉② r❛ λ = ❑❤✐ ➤ã [a, b]A = 0, ∀a ∈ A, ➤✐Ò✉ ♥➭② s✉② r❛ a ∈ Z(A) ❍➡♥ ♥÷❛ T (δ(a), b) = 0, ∀b ∈ A ♥➟♥ δ(a) = ♠➭ δ ❦❤➯ ♥❣❤Þ❝❤ tr➟♥ Z(A) ♥➟♥ a = ❉♦ ✈❐②✱ X = αe∗ ❉♦ ➤ã✱ Z(G) ⊆ Ke∗ ❱❐② Z(G) = Ke∗ ✈➭ G ❧➭ ➤➵✐ sè ▲✐❡ ❣✐➯✐ ➤➢ỵ❝ ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧ý ✷✳✷✳✹ s✉② r❛ G ❧➭ ➤➵✐ sè ▲✐❡ ➤Þ❛ ♣❤➢➡♥❣✳ ✐✐✮ ●✐➯ sư G ❧➭ ➤➵✐ sè ▲✐❡ ➤Þ❛ ♣❤➢➡♥❣✳ ❚õ ➜Þ♥❤ ❧ý ✷✳✷✳✹✱ t❛ ❝ã ❤➵♥❣ tư ▲❡✈✐ ❝đ❛ ♥ã ❧➭ ❤➵♥❣ tư ➤➡♥✳ ➜✐Ị✉ ♥➭② ❝ã ♥❣❤Ü❛ ❝➝♥ ❤➵♥❣ tư ▲❡✈✐ ❝đ❛ q✉❛ ♣❤Ð♣ ❝❤✐Õ✉ G, t❤❡♦ ❬✾❪ t❤× G R(G) ❧➭ ✐❞❡❛❧ ❝ù❝ ➤➵✐ ❝ñ❛ G ❚❤❐t ✈❐②✱ ♥Õ✉ S ❧➭ ♠ë ré♥❣ ❦Ð♣ ❝ñ❛ A = R(G)/(R(G))⊥ ❜ë✐ ❧➭ S ψ : S Dera (A, T ) ợ ị (X)(N (Y )) = N ([X, Y ]), ∀X ∈ S; Y ∈ R(G) ✈➭ N : R(G) −→ A ❧➭ ♠ét ♣❤Ð♣ ❝❤✐Õ✉ ❝❤Ý♥❤ t➽❝✳ ❱× G ❧➭ ➤➵✐ sè ▲✐❡ ➤➬② ➤đ✱ ♥➟♥ R(G) ❧➭ ✐❞❡❛❧ ❧ị② ❧✐♥❤✱ ❞♦ ✈❐② A ❝ò♥❣ ❧➭ ➤➵✐ sè ▲✐❡ ❧ò② ❧✐♥❤✳ ❈❤ø♥❣ ♠✐♥❤ (Z(A))S = {0} ⇔ ψ(X)/N (Y ) = 0, ∀X ∈ S; Y ∈ R(G) ❚r➢í❝ ❤Õt t❛ ❝❤ø♥❣ ♠✐♥❤ (R(G))⊥ = Z(R(G)) ❧➭ t➞♠ ❝đ❛ R(G) ❱× B ❜✃t ❜✐Õ♥ ✈➭ ❈❤➢➡♥❣ ✷✳ ✹✵ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ [G, R(G)] = R(G) ♥➟♥ Z(R(G)) ⊂ (R(G))⊥ ❍➡♥ ♥÷❛✱ B((R(G))⊥ , R(G)) = B((R(G))⊥ , [G, R(G)]) = B([(R(G))⊥ , R(G)], G) = {0} ✭✈× B ❜✃t ❜✐Õ♥✮✱ ➤✐Ị✉ ♥➭② ❝❤♦ t❛ t❤✃② r➺♥❣ [(R(G))⊥ , R(G)] = {0}, ❦❤✐ ➤ã (R(G))⊥ ⊂ Z(R(G)) ❱❐② (R(G))⊥ = Z(R(G)) ❤❛② A = R(G)/Z(R(G)) ➜Ó ❝❤ø♥❣ ♠✐♥❤ (Z(A))S = {0}, t❛ ❧✃② Y ∈ R(G) s❛♦ ❝❤♦ N (Y ) ∈ Z(A) ✈➭ N ([X, Y ]) = 0, ∀X ∈ S ➤✐Ò✉ ♥➭② ❝❤ø♥❣ tá ❚❛ ❝ã [Y, R(G)] ⊂ Z(R(G)) ✈➭ [Y, S] ⊂ Z(R(G)) [Y, G] ⊂ Z(R(G)) ❉♦ ➤ã B(R(G), [Y, G]) = B([Y, R(G)], G) = {0} s✉② r❛ [Y, R(G)] ✈➭ = ❤❛② Y ∈ Z(R(G)) ❉♦ ✈❐②✱ N (Y ) = 0, ✈× A = R(G)/Z(R(G)) N : R(G) −→ A ❱❐② (Z(A))S = {0} ◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sö (A, T ) (G, B) ❜ë✐ ➤➵✐ sè ➤➡♥ S ❧➵✐ ♠ë ré♥❣ ❦Ð♣ ❝ñ❛ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❧ị② ❧✐♥❤ q✉❛ ♣❤Ð♣ ❜✐Ĩ✉ ❞✐Ơ♥ ψ : S −→ Dera (A, T ) (Z(A))S = {0} ❑❤✐ ➤ã✱ G = S ∗ ⊕ A ⊕ S, ✈í✐ S ❧➭ ✐❞❡❛❧ ❣✐➯✐ ➤➢ỵ❝ ❝đ❛ ❝ã G ❧➭ ➤➵✐ sè ➤➡♥ ❝ñ❛ G t❤á❛ ♠➲♥ ✈➭ S∗ ⊕ A ✈× ♥ã ❧➭ ♠ë ré♥❣ t➞♠ ❝đ❛ ♠ét ➤➵✐ sè ▲✐❡ ❧ị② ❧✐♥❤✳ ❈❤ó♥❣ t❛ R(G) = S ∗ ⊕ A ✈➭ S ❧➭ ❤➵♥❣ tư ▲❡✈✐ ❝đ❛ G ❇➞② ❣✐ê t❛ ❝❤ø♥❣ ♠✐♥❤ G ❧➭ ➤➬② ➤ñ Z(G) = {0} f ∈ S ∗, ✈➭ ➤Þ❛ ♣❤➢➡♥❣✳ ➜Ĩ ❝❤ø♥❣ ♠✐♥❤ a ∈ A, s ∈ S G ➤➬② ➤ñ t❛ ❝❤ø♥❣ ♠✐♥❤ s❛♦ ❝❤♦ X = f + a + s ∈ Z(G) ❚❤❐t ✈❐②✱ ①Ðt ❑❤✐ ➤ã✱ ✈í✐ s ∈ S = [s , X] = [s , f + a + s] = −f.adS s∗ + ψ(s )(a) + [s , s]S ❞➱♥ ➤Õ♥ f.adS s = 0, [s , s] = 0, ∀s ∈ S ❉♦ ➤ã✱ f = 0; s = ❱× S ♥➭② ❝ã ♥❣❤Ü❛ a ∈ (Z(A))S ❇➞② ❣✐ê ✈í✐ a ∈ A t❛ ❝ã✿ ϕ(a , a) ∈ S ∗ ➜✐Ò✉ ♥➭② ❧➭ ➤➡♥✱ ➤✐Ò✉ = [a , X] = [a , f + a + s] = ϕ(a , a) + [a , a]A ✈í✐ t❤× ợ ị (a , a)(s ) = B((s )(a ), a) = −B(a , ψ(s )(a)), ∀s ∈ S ❈❤➢➡♥❣ ✷✳ ❱× a ∈ (Z(A))S ◆➟♥ a ❱❐② G ✹✶ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ ♥➟♥ ψ(s )(a) = 0, ✈❐② ϕ(a , a) = ❤❛② [a , a]A = 0, ∀a ∈ A ∈ Z(A) ❤❛② a ∈ Z(A) ∩ (Z(A))S = {0} ➜✐Ò✉ ♥➭② ❝ã ♥❣❤Ü❛ Z(G) = {0} ❧➭ ➤➵✐ sè ▲✐❡ ➤➬② ➤đ✳ ❍➡♥ ♥ị❛✱ Soc(G) = (R(G))⊥ ♠ét ✐❞❡❛❧ ➤➡♥✱ ❞♦ ➤ã ✷✳✷✳✻✳ ◆❤❐♥ ①Ðt✳ G ❧➭ ♠ét ✐❞❡❛❧ ❝ù❝ t✐Ĩ✉✱ ✈× ❤➵♥❣ tư ▲❡✈✐ S ❝đ❛ G ❧➭ ❧➭ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤Þ❛ ♣❤➢➡♥❣✳ ✰✮ ❚õ ❦Õt q✉➯ ❝đ❛ ➤Þ♥❤ ❧ý tr➟♥ t❛ t❤✃② r➺♥❣ ➤Ĩ ♥❣❤✐➟♥ ❝ø✉ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ✈í✐ sè ❝❤✐Ị✉ ❜➺♥❣ ✷ t❛ ❝❤ó ý tí✐ ❤❛✐ tr➢ê♥❣ ❤ỵ♣ ❧➭ ➤➵✐ sè ▲✐❡ ❣✐➯✐ ➤➢ỵ❝ ✈➭ ➤➵✐ sè ▲✐❡ ➤➬② ➤đ✳ ✰✮ ❈❤ó♥❣ t❛ ❜✐Õt ♠ä✐ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❜✃t ❦❤➯ q✉② s❛♦ ❝❤♦ dq (G) = ❧➭ ♠ét ➤➵✐ sè ▲✐❡ ➤Þ❛ ♣❤➢➡♥❣ ♥❤➢♥❣ ợ ú ự tế ó ữ số ▲✐❡ ➤Þ❛ ♣❤➢➡♥❣ ❝ã ❝❤✐Ị✉ t♦➭♥ ♣❤➢➡♥❣ ❧í♥ ❤➡♥ ✷✳ ❈❤➢➡♥❣ ✷✳ ✷✳✸ ✹✷ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ ▼ét sè ❧í♣ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷✳ ◆é✐ ❞✉♥❣ ❝❤Ý♥❤ ❝đ❛ ♣❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝đ❛ ♠ét sè ❧í♣ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❝ã ❝❤✐Ị✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷✱ ❝ơ t❤Ĩ ♥❤➢ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❧ò② ❧✐♥❤✱ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤➬② ➤đ✱✳✳✳ ✷✳✸✳✶✳ ➜Þ♥❤ ❧ý✳✭①❡♠ ❬✸❪✮ ❈❤♦ (A, T ) ❧➭ ♠ét ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ✈➭ δ ∈ Dera (A, T ) ❧➭ ♠ét t♦➳♥ tö ❦❤➯ ♥❣❤Þ❝❤✳ ❑❤✐ ➤ã ♠ë ré♥❣ ❦Ð♣ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ❈❤ø♥❣ ♠✐♥❤✳ ❱× (G, B) ❝đ❛ (A, T ) ❜ë✐ δ ❝ã ❝❤✐Ò✉ (G, B) ❧➭ ♠ë ré♥❣ ❦Ð♣ ❝ñ❛ (A, T ) ❜ë✐ δ ♥➟♥ G = Ke∗ ⊕ A ⊕ Ke ✭✈í✐ Ke∗ = Z(G)) ➜➷t α = K(e, e), β = K(e, e∗ ) ❑❤✐ ➤ã ✈í✐ ♠ä✐ x ∈ A, t❛ ❝ã ●ä✐ K ❧➭ ♠ét ❞➵♥❣ s♦♥❣ t✉②Õ♥ tÝ♥❤ ➤è✐ ①ø♥❣ ❜✃t ❜✐Õ♥ tr➟♥ G K(e, x) = K(e, [e, δ −1 x]) = K([e, e], δ −1 x) = 0( ❞♦ tÝ♥❤ ❝❤✃t ❜✃t ❜✐Õ♥ ❝ñ❛ K) ❚➢➡♥❣ tù K(e∗ , x) = K(e∗ , [e∗ , δ −1 x]) = ✈× Ke∗ = Z(G) ✭t❤❡♦ ➜Þ♥❤ ❧ý ✷✳✷✳✺✮✳ ❇➞② ❣✐ê✱ ♥Õ✉ x, y ∈ A s❛♦ ❝❤♦ T (x, y) = t❤× = K([δ −1 x, y], e∗ ) = K([δ −1 x, y]A, e∗ )+T (x, y)K(e∗ , e∗ ) = T (x, y)K(e∗ , e∗ ) ❚õ ➤➞② s✉② r❛ K(e∗ , e∗ ) = ❤➡♥ ♥÷❛✱ ✈í✐ ♠ä✐ x, y ∈ A ❝❤ó♥❣ t❛ ❝ã✿ K(x, y) = K(x, [e, δ −1 y]) = K([δ −1 y, x], e) = T (x, y)K(e∗ , e) = βT (x, y) = βB(x, y) ✭✈× (G, B) ❧➭ ♠ë ré♥❣ ❦Ð♣ ❝ñ❛ (A, T ) ♥➟♥ B(x, y) = T (x, y), ∀x, y ∈ A✮✳ ❉♦ ✈❐②✱ ♥Õ✉ S ❧➭ ❞➵♥❣ s♦♥❣ t✉②Õ♥ tÝ♥❤ ➤➢ỵ❝ ①➳❝ ị tì ễ ứ ợ r S(e, e) = 1, S(Ke∗ ⊕ A, G) = K = αS + βB, ❞♦ ➤ã dimF(G) = ◗✉❛ ➤Þ♥❤ ❧ý tr➟♥ t❛ t❤✃② ➤✐Ị✉ ❦✐Ư♥ ❦❤➯ ♥❣❤Þ❝❤ ❝đ❛ ✈✐ ♣❤➞♥ ➤✐Ị✉ ❦✐Ư♥ ❦❤➯ ♥❣❤Þ❝❤ tèt ❤➡♥ tr➟♥ ❝➯ ➤➢ỵ❝ ①❡♠ ♥❤➢ t➢➡♥❣ ➤➢➡♥❣ ✈í✐ [A, A] ✈➭ Z(A) U = [A, A] + Z(A) δ ❝ã t❤Ó t❤❛② t❤Õ ❜ë✐ ❚❤ù❝ ❝❤✃t ❤❛✐ ➤✐Ị✉ ❦✐Ư♥ ♥➭② ❈❤➢➡♥❣ ✷✳ ✹✸ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ ●✐➯ sö ❧➭ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ✈➭ (A, T ) ❝♦♥ t✉②Õ♥ tÝ♥❤ δ ∈ Dera (A, T ) ❳Ðt ❦❤➠♥❣ ❣✐❛♥ K × K × A × End(A) : ε(A, T, δ) = (α, λ, m, l) ∈ K × K × A × Ker(δ)|δl = lδ = αδ + adA (m) ✷✳✸✳✷✳ ➜Þ♥❤ ❧ý✳ ✭①❡♠ ❬✸❪✮ ●ä✐ (G, B) ❧➭ ♠ë ré♥❣ ❦Ð♣ ❝ñ❛ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ (A, T ) ❜ë✐ δ, ❣✐➯ sö Z(G) = Ke∗ ❑❤✐ ➤ã✿ ✐✮ ◆Õ✉ (α, λ, m, l) ∈ ε(A, T, δ) t❤× tù ➤å♥❣ ❝✃✉ tế tí D(,,m,l) ủ G ợ ị ĩ D(,,m,l) (e∗ ) = ([αe∗ , x], e) D(α,λ,m,l) (e) = ([αe∗ + m + αe, x], e) D(α,λ,m,l) (a) = l(a) + T (m, a)e∗ , ∀a ∈ A ❧➭ ♠ét ♣❤➬♥ tö tr♦♥❣ CentS (G, B) ✐✐✮ ❙ù t➢➡♥❣ ø♥❣ φ :(A, T, δ) −→ CentS (G, B) (α, λ, m, l) −→ D(α,λ,m,l) ❧➭ ♠ét ➳♥❤ ①➵ ➤➻♥❣ ❝✃✉ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡✳ ✐✐✐✮ ➜✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ ➤Ó dq (G) = ❧➭ ε(A, T, δ) = {(α, λ, 0, l)|α, λ ∈ K} ✷✳✸✳✸✳ ➜Þ♥❤ ❧ý✳ ❈❤♦ (A, T ) ❧➭ ♠ét ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❧ò② ❧✐♥❤ ✈➭ δ ∈ Dera (A, T ) ❧➭ ♠ét t♦➳♥ tư ❦❤➯ ♥❣❤Þ❝❤ tr➟♥ A ➤è✐ ✈í✐ ❑❤✐ ➤ã δ n ✈➭ Z(A) ❳Ðt A = K ⊕ C ❧➭ sè ♥❣✉②➟♥ ❞➢➡♥❣ s❛♦ ❝❤♦ K = C ⊥ , [K, K] ⊂ K ♣❤Ð♣ ❝❤✐Õ✉ ❝ñ❛ A ✈➭♦ K ✈➭ K = Ker(δ n ) [K, C] C ợ ị ĩ trể tí ợ ủ ữ ế C = n (A) θ : A −→ K θ(k = c) = k.∀k ∈ K, c ∈ C ❧➭ t❤× θ([C, C]) = K ❈❤ø♥❣ ♠✐♥❤✳ T ♥➟♥ ❚❤❡♦ tr➟♥ t❛ ❝ã δ(K) ⊂ K ✈➭ δ(C) ⊂ C ❱× δ ❧➭ ➤è✐ ①ø♥❣ ❧Ư❝❤ ✈í✐ K = C ⊥ ữ [K, K] K, ì ọ x, y ∈ K, t❛ ❝ã✿ 2n 2n δ [x, y] = k=0 2n k [δ x, δ 2n−k y] = ⇒ [x, y] ∈ Ker(δ 2n ) = Ker(δ n ) k ❈❤➢➡♥❣ ✷✳ ❉♦ ➤ã✱ [K, C] ⊂ C ✈× ❚✐Õ♣ t❤❡♦✱ t❛ ①Ðt k ∈ K ❧✃② ✹✹ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ ✈➭ T ❜✃t ❜✐Õ♥ ✈➭ K = C ⊥ I = θ([C, C]) x, y ∈ C ◆Õ✉ ✈➭ ❝❤Ø r❛ r➺♥❣ [x, y] = k1 + c1 I ❧➭ ✐❞❡❛❧ ❝đ❛ ✭✈í✐ K ❚❤❐t ✈❐②✱ (k1 , c1 ) ∈ K × C ✮ t❤× θ([x, y]) = [x, y] − c1 ◆➟♥ [k, θ([x, y])] = [k, [x, y]] − [k, c1 ] = [x, [k, y]] + [y, [x, k]] + [c1 , k] ❱× [K, C] ⊂ C, ♥➟♥ [k, θ([x, y])] = θ[x, [k, y]] + [y, [x, k]] ∈ I ❙✉② r❛ I ❝đ❛ K ❇➞② ❣✐ê✱ t❛ ①Ðt t❐♣ ❤ỵ♣ J = I ⊥ ∩ K, tr♦♥❣ ➤ã I ⊥ T ❱× T ❜✃t ❜✐Õ♥ ♥➟♥ J ❧➭ ✐❞❡❛❧ ❝ñ❛ ❧➭ trù❝ ❣✐❛♦ ❝đ❛ I ❧➭ ✐❞❡❛❧ ➤è✐ ✈í✐ K ❍➡♥ ♥÷❛ T ([J, C], C) = T (J, [C, C]) = T (J, I) = {0} ✈× T (K, C) = {0}, ❉♦ ➤ã✱ J ♥➟♥ ❧➭ ✐❞❡❛❧ ❝ñ❛ [J, C] = {0} ✷✳✸✳✹✳ ➜Þ♥❤ ❧ý✳ ●✐➯ sư δ tr➟♥ ✈➭ C = δ n (A) ✈➭ T |C×C ❦❤➠♥❣ s✉② ❜✐Õ♥✳ (G, B) Z(A) ❱❐② J = {0} ♥➟♥ K = I = θ([C, C]) ❧➭ ♠ë ré♥❣ ❦Ð♣ ❝ñ❛ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❧ò② ❧✐♥❤ (A, T ) ❜ë✐ δ ◆Õ✉ dimF(G) = t❤× δ ➜➷t A [K, C] ⊂ C A ◆Õ✉ J = {0} t❤× J ∩ Z(A) = {0}, ➤✐Ị✉ ♥➭② ♠➞✉ t❤✉❐♥ ✈í✐ tí ị ủ ứ ì ị tr ZA ([A, A]) = K ⊕C ❧➭ ❦❤❛✐ tr✐Ó♥ t❤Ý❝❤ ❤ỵ♣ ❝đ❛ A, tr♦♥❣ ➤ã K = Ker(δ n ) ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ m ∈ ZA ([A, A]) ∩ Ker(δ) δ ❑❤✐ ➤ã ❦❤➠♥❣ ❦❤➯ ♥❣❤Þ❝❤ tr➟♥ m ∈ Z(K) ✈× ZA ([A, A]) ✈➭ t❛ ❧✃② CA ([A, A]) ⊂ ZA (K) ●ä✐ l : A −→ A tế tí ợ ị l(k + c) = [m, δ n−1 (a)], ∀k ∈ K, ∀c = δ(a) ∈ δ(A) ❚❛ t❤✃② ➳♥❤ ①➵ ♥Õ✉ l ợ ị ĩ tr ụ tộ ♣❤➬♥ tư ➤➵✐ ❞✐Ư♥✱ ✈× c = δ(a) = δ(a1 ) t❤× t❛ ❝ã✿ (a − a1 ) ∈ K ⇒ δ n−1 (a − a1 ) ∈ K ❱× m ∈ Z(K) ♥➟♥ [m, δ n−1 (a − a1 )] = ❈❤➢➡♥❣ ✷✳ ✹✺ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ ▼➷t ❦❤➳❝✱ K ✈➭ C ỉ♥ ➤Þ♥❤ ❜ë✐ δ ✱ ♥➟♥ ∀k ∈ K ✈➭ ∀c = δ n (a) ∈ C, t❛ ❝ã✿ lδ(k + c) = l(δ(k) + δ(c)) = l(δ n+1 (a)) = [m, δ n (a)] = [m, c] = [m, k + c] δl(k + c) = δ[m, δ n−1 (a)] = [δ(m), δ n−1 (a)] + [m, δ n (a)] = [m, δ n (a)] = [m, k + c] ❙✉② r❛ lδ = δl = adA (m) ữ ì l ố ứ ì t ể tr ➤✐Ị✉ ➤ã ✈í✐ δ(m) = ♥❤❐♥ ➤➢ỵ❝ T (l(k + δ n (a)), k1 + δ n (a1 )) = T ([m, δ n−1 (a)], δ n (a1 )) = −T (δ[m, δ n−1 (a)], δ n−1 (a1 )) = T k + δ n (a), l(k + δ n (a1 )) l[x, y] = [lx, y] = 0, ∀x, y ∈ A ❇➞② ❣✐ê t❛ ❝❤ø♥❣ ♠✐♥❤ ➤è✐ ✈í✐ T ♥➟♥ ➤đ ➤Ĩ ❦Õt ❧✉❐♥ k ∈ K, c = δ n (a) ∈ C ✈➭ [lx, y] = 0, ∀x, y ∈ A ❚❤❐t ✈❐②✱ ✈× ●✐➯ sư l ➤è✐ ①ø♥❣ x = k + c, ✈í✐ y, z ∈ A, t❛ ❝ã✿ T ([lx, y], z) = T (lx, [y, z]) = T ([m, δ n−1 (a)], [y, z]) = T m, δ n−1 (a), [y, z] = ⇒ [lx, y] = ◆➟♥ = m ∈ Ker(δ) ✈➭ l ∈ CentS (A, T ) ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉❐♥ ✈í✐ ➜Þ♥❤ ❧ý ✷✳✸✳✷ ❧➭ ✷✳✸✳✺✳ ➜Þ♥❤ ❧ý✳ ●✐➯ sư (G, B) ➤ã δ (⇒) ì ị tr dq (A) ❦❤❛✐ tr✐Ĩ♥ t❤Ý❝❤ ❤ỵ♣ ❝đ❛ K = Ker(δ n ) ZA (δ n−1 ([A, A])) ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ δ ❈❤ø♥❣ ♠✐♥❤✳ (0, 0, m, l) ∈ ε(A, T, δ) ❧➭ ♠ë ré♥❣ ❦Ð♣ ❝ñ❛ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❧ò② ❧✐♥❤ (A, T ) ❜ë✐ δ ➜➷t A = K ⊕ C ♥❣✉②➟♥ ❞➢➡♥❣ s❛♦ ❝❤♦ s❛♦ ❝❤♦ ❜➯♦ t♦➭♥ ✈➭ C = δ n (A) ❦❤➯ ♥❣❤Þ❝❤ tr➟♥ [A, A] ♥➟♥ A ➤è✐ ✈í✐ δ ❑❤✐ ➤ã δ ✈➭ n ❧➭ sè ❦❤➯ ♥❣❤Þ❝❤ tr➟♥ ZA ([A, A]) ZA ([A, A]) ⊆ ZA (δ n−1 ([A, A])) ZA ( n1 ([A, A])) tì ó ị tr ZA ([A, A]) ❉♦ ❈❤➢➡♥❣ ✷✳ ✹✻ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ị✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ (⇐) ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ δ ❦❤➠♥❣ ❦❤➯ ♥❣❤Þ❝❤ tr➟♥ ZA (δ n−1 ([A, A])) ❳Ðt✿ m ∈ ZA (δ n−1 ([A, A])) ∩ Ker(δ) ỗ k K; x, y C δ n (a) tå♥ t➵✐ a, a1 ∈ C s❛♦ = x, δ n (a1 ) = y ❑❤✐ ➤ã T ([m, k], θ[δ n (a), δ n (a1 )]) = T ([m, k], [δ n (a), δ n (a1 )]) = −T (k, [m, [δ n (a), δ n (a1 )]]) = T (k, [δ n (a), [δ n (a1 ), m]]) + T (k, [δ n (a1 ), [m, δ n (a)]]) = (−1)n T ([δ n ([δ n (a), k]), m], a1 ) + (−1)n T ([δ n ([k, δ n (a1 )]), m]) = ✈× δ[A, A] ⊂ [A, A] ✈➭ m ∈ ZA (δ n−1 ([A, A])) ▼➷t ❦❤➳❝✱ K = θ([C, C]) ✈➭ T |K×K ⇒ m ∈ Z(K) ⇒ m ∈ Z(Ker(δ)) ♥➟♥ δ n−1 ([m, [x, y]]) ❦❤➠♥❣ s✉② ❜✐Õ♥ ♥➟♥ [m, k] = m ∈ ZA (δ n−1 ([A, A])) ∩ Ker(δ) ❱× = [m, δ n−1 ([x, y])] = 0, ∀x, y ∈ A ⇒ [m, [x, y]] ∈ Ker(δ n−1 ) = K ◆ã✐ ❝➳❝❤ ❦❤➳❝✱ ♥Õ✉ x, y ∈ A ✈➭ [x, y] = k + c ✈í✐ k ∈ K, c ∈ C ❱❐② ✈➭ m ∈ Z(K), ❦❤✐ ➤ã [m, [x, y]] = [m, c] ∈ [K, C] ⊂ C [m, [x, y]] ∈ [K, C] = {0} ⇒ m ∈ ZA ([A, A]) ∩ Ker(δ) ❚❤❡♦ ➤✐♥❤ ❧ý ✷✳✸✳✹ t❛ t❤✃② ➤✐Ị✉ ❦✐Ư♥ ủ ể ợ tr ị ý ❦❤➠♥❣ ❝➬♥ t❤✐Õt✳ ❱Ý ❞ơ sù ♠ë ré♥❣ ❝đ❛ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❣✐➯✐ ➤➢ỵ❝ ✈í✐ (G, B) dq (G) = ✈➭ ë ➤ã (G, B) ❧➭ ♠ë ré♥❣ ❦Ð♣ ❝đ❛ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❧ị② ❧✐♥❤ (A, T ) ❜ë✐ δ ∈ Dera (A, T ) ❧➭ ❦❤➠♥❣ ❦❤➯ ♥❣❤Þ❝❤ tr➟♥ t♦➭♥ ❜é A ✷✳✸✳✻✳ ❍Ư q✉➯✳ ●✐➯ sö ❜✐Õ♥ ✈➭ V δ ❧➭ tù ➤å♥❣ ❝✃✉ ♣❤➯♥ ①ø♥❣ ❝đ❛ ỉ♥ ➤Þ♥❤ ❜ë✐ t❤Ĩ (V, B) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ ❝ã ❞➵♥❣ s♦♥❣ t✉②Õ♥ tÝ♥❤ ❦❤➠♥❣ s✉② δ U ⊥ ⊂ U ◆Õ✉ δ ✈➭ V ●ä✐ U ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ t✉②Õ♥ tÝ♥❤ ❝ñ❛ ❦❤➯ ị tr U tì ị tr t V ❱× ❈❤ø♥❣ ♠✐♥❤✳ ❣✐❛♥ ❝♦♥ W ❝đ❛ ❝❤Ø ❝ã t❤Ĩ ♥Õ✉ ❳Ðt ✈❡❝t➡ U⊥ V ❤♦➭♥ t♦➭♥ ➤➻♥❣ ❤➢í♥❣ ✈➭ s❛♦ ❝❤♦ V = U ⊕W B ❦❤➠♥❣ s✉② ❜✐Õ♥ ♥➟♥ tå♥ t➵✐ ❦❤➠♥❣ ✈➭ t❤á❛ ♠➲♥ B(w, U ⊥ ) = {0}, w ∈ W w = v = u+w ∈ V ✈í✐ u ∈ U, w ∈ W δ(w) + δ(u) = ⇒ δ(w) = −δ(u) ∈ U ✈➭ ❣✐➯ sư ▼➷t ❦❤➳❝ ✈× δ δ(v) = ❧➭ ♣❤➯♥ ①ø♥❣ ✈➭ ❞♦ ➤ã U æ♥ ❈❤➢➡♥❣ ✷✳ ➤Þ♥❤ ❜ë✐ t❛ ❝ã ✹✼ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ δ ♥➟♥ U ⊥ ❝ị♥❣ ỉ♥ ➤Þ♥❤ ❜ë✐ δ ❉♦ ➤ã δ(U ⊥ ) = U ó ỗ a U ⊥ B(δ(a), w) = −B(a, δ(w)) = ➜✐Ò✉ ♥➭② ❝❤ø♥❣ tá w trù❝ ❣✐❛♦ ✈í✐ U ⊥ ♥➟♥ w = ✷✳✸✳✼✳ ❍Ư q✉➯✳ ●✐➯ sư ❦❤➯ ♥❣❤Þ❝❤ tr➟♥ (A, T ) ❧➭ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❧ò② ❧✐♥❤ ✈➭ δ ∈ Dera (A, T ) Z(A) ❳Ðt A = K ⊕ C ❧➭ ❦❤❛✐ tr✐Ó♥ t❤Ý❝❤ ❤ỵ♣ ❝đ❛ A ➤è✐ ✈í✐ δ ✈➭ n ❧➭ sè ♥❣✉②➟♥ ❞➢➡♥❣ s❛♦ ❝❤♦ K = Ker(δ n ) ✈➭ C = δ n (A) ❱✐ ♣❤➞♥ δ ❦❤➯ ♥❣❤Þ❝❤ tr➟♥ A ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ C ❱Ý ❞ô ✹✳ ●ä✐ ❧➭ ➤➵✐ sè ❝♦♥ ❝ñ❛ A A ❧➭ ❦❤➠♥❣ ❣✐❛♥ t✉②Õ♥ tÝ♥❤ ➤➢ỵ❝ t➵♦ ❜ë✐ ✽ ✈❡❝t➡ {xi , fi |1 ≤ i ≤ 4} ❧➭ ♠ét ➤➵✐ sè tí ợ ị s [x1 , xi ] = xi+1 , ≤ i ≤ [x1 , fi ] = −fi−1 , ≤ i ≤ [xi , fi+1 ] = fi , ≤ i ≤ ✈➭ ❣ä✐ T ❧➭ ♠ét s tế tí tr A ợ ị T (xi , xj ) = T (fi , fj ) = 0, ∀i, j T (xi , fj ) = 0, ∀i = j T (xi , fi ) = 1, ∀i ≤ ❑❤✐ ➤ã✱ ❝➷♣ δ ❝ñ❛ (A, T ) ❧➭ ♠ét ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❧ị② ❧✐♥❤✳ ❚ù ➤å♥❣ ❝✃✉ t✉②Õ♥ tÝ♥❤ A ➤➢ỵ❝ ①➳❝ ➤Þ♥❤ ♥❤➢ s❛✉ δ(x1 ) = x1 , δ(x2 ) = −x2 , δ(x3 ) = 0, δ(x4 ) = x4 δ(f1 ) = −f1 , δ(f2 ) = f2 , δ(f3 ) = 0, δ(f4 ) = −f4 ❧➭ ♠ét ➤➵♦ ❤➭♠ ♣❤➯♥ ①ø♥❣ ❝đ❛ ❱× (A, T ) Z(A) ➤➢ỵ❝ t➵♦ ❜ë✐ {x4 , f1 } ♥➟♥ râ r ị tr Z(A) ì Z(A) ker() = {0} ❚✉② ♥❤✐➟♥ Z2 (A) ∩ ker(δ) = {0}, ✈× x3 , f3 ∈ Z2 (A) ∩ ker(δ), ❞♦ ị tr Z2 (A) ì ♠ë ré♥❣ ❦Ð♣ (G, B) ❝ñ❛ (A, T ) ❜ë✐ δ ❧➭ ♠ét ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤Þ❛ ♣❤➢➡♥❣ ♥❤➢♥❣ ❝ã sè ❝❤✐Ò✉ dq (G, B) > ❈❤➢➡♥❣ ✷✳ ✹✽ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ ❚❤❡♦ ➜Þ♥❤ ❧ý ✷✳✷✳✺✱ t❛ ❝ã ♠ë ré♥❣ ❦Ð♣ ❝đ❛ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❧ị② ❧✐♥❤ ❜ë✐ ♠ét ➤➵✐ sè ▲✐❡ ➤➡♥ ❧➭ ♠ét ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤➬② ➤đ✳ ❚✐Õ♣ t❤❡♦ tr♦♥❣ ♣❤➬♥ ♥➭② ❝❤ó♥❣ t❛ sÏ ♥❣❤✐➟♥ ❝ø✉ ♠ét sè tÝ♥❤ ❝❤✃t ❝ñ❛ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤➬② ➤đ ❝ã ❝❤✐Ị✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷✳ ✷✳✸✳✽✳ ➜Þ♥❤ ❧ý✳ ✭①❡♠ ❬✸❪✮ ▼ét ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❧➭ ➤➬② ➤ñ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ t➞♠ ❝đ❛ ♥ã ❜➺♥❣ ✵✳ ✷✳✸✳✾✳ ➜Þ♥❤ ❧ý✳ ✭①❡♠ ❬✸❪✮ ❈❤♦ ➤➡♥✳ ●✐➯ sư tå♥ t➵✐ ❜✐Ĩ✉ ❞✐Ơ♥ ❦Ð♣ G = S∗ ⊕ A ⊕ S ❝ñ❛ (A, T ) ❧➭ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ✈➭ ψ : S −→ Dera (A, T ) (A, T ) ❜ë✐ S t❤❡♦ ψ ●ä✐ S (G, B) ➤➵✐ sè ▲✐❡ ❧➭ ♠ë ré♥❣ ❳Ðt ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ t✉②Õ♥ tÝ♥❤ E = K × K × HomS (S, A) × CentS (A, T ) ➤➢ỵ❝ ❝❤♦ ❜ë✐✿ ε(A, T, S, ψ) = {(α, λ, F, l) ∈ E|l◦ψ(x) = ψ(x)◦l = αψ(x)a dA F (x), ∀x ∈ S} ❑❤✐ ➤ã t❛ ❝ã✿ ✐✮ ◆Õ✉ (α, λ, F, l) ∈ ε(A, T, S, ) tì tự tế tí D(,,F,l) ợ ➤Þ♥❤ ❜ë✐✿ D(α,λ,F,l) (f ) = αf, ∀f ∈ S ∗ D(α,λ,F,l) (a) = T (a, F (.)) + l(a), ∀a ∈ A D(α,λ,F,l) (x) = λκ(x, ) + F (x) + αx, ∀x ∈ S tr♦♥❣ ➤ã κ ❦ý ❤✐Ư✉ ❧➭ ❞➵♥❣ ❑✐❧❧✐♥❣ ❝đ❛ G, ❧➭ ♠ét ♣❤➬♥ tư tr♦♥❣ CentS (G, B) ✐✐✮ ❙ù t➢➡♥❣ ø♥❣ φ : ε(A, T, S, ψ) −→ CentS (G, B) (α, λ, F, l) −→ D(α,λ,F,l) ❧➭ ♠ét ➳♥❤ ①➵ ➤➻♥❣ ❝✃✉ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡✳ ✐✐✐✮ ➜✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ ➤Ĩ dq (G) = ❧➭ ε(A, T, S, ψ) = {(α, λ, 0, αidA )|α, λ ∈ K} ❙❛✉ ➤➞② t❛ ➤➢❛ r❛ ♠ét sè ✈Ý ❞ơ ❝❤ø♥❣ tá ❧í♣ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤➬② ➤ñ ❧✉➠♥ tå♥ t➵✐✳ ❈❤➢➡♥❣ ✷✳ ❱Ý ❞ơ ✺✳ ❈❤♦ S ♣❤Ð♣ ❜✐Ĩ✉ ❞✐Ơ♥ ❱Ý ❞ơ ✻✳ ✹✾ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ị✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷ ❈❤♦ ❧➭ ➤➵✐ sè ▲✐❡ ➤➡♥✱ ❦❤✐ ➤ã ♠ë ré♥❣ ❦Ð♣ ❝ñ❛ {0} q✉❛ ψ = ❝ã sè ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ✷✳ S = sl(2) ✈➭ A ❧➭ sl(2)− ♠♦❞✉❧❡ ❜✃t ❦❤➯ q✉② ❦❤➳❝ dim(A) = 2k + 1, k ≥ ❱× A ➤➻♥❣ ❝✃✉ ✈í✐ A∗ ➤è✐ ♥❣➱✉ ❝đ❛ G = S ⊕ S∗ A✮✳ ✭tr♦♥❣ ➤ã A∗ ❧➭ s❛♦ ❝❤♦ sl(2)− ♠♦❞✉❧❡ ◆Õ✉ tå♥ t➵✐ ♠ét ❞➵♥❣ s♦♥❣ t✉②Õ♥ tÝ♥❤ ➤è✐ ①ø♥❣ ❦❤➠♥❣ s✉② ❜✐Õ♥ T : A × A −→ K ✈➭ T ❝ò♥❣ ❧➭ sl(2)− ❜✃t ❜✐Õ♥✱ ❝ã ♥❣❤Ü❛ ❧➭ T (x, a, a1 ) = −T (a, x, a1 ), ∀x ∈ sl(2); a, a1 ∈ A ❱× A ❝ñ❛ ❧➭ ➤➵✐ sè ▲✐❡ ❣✐❛♦ ❤♦➳♥ ✈➭ ➳♥❤ ①➵ ψ : sl(2) −→ gl(A) sl(2) ❦Õt ❤ỵ♣ ✈í✐ sl(2) − module tr♦♥❣ A ❑❤✐ ➤ã✱ ♥Õ✉ G (A, T ) ❜ë✐ sl(2) q✉❛ ♣❤Ð♣ ❜✐Ĩ✉ ❞✐Ơ♥ ψ ❍➡♥ ữ ế tì é ể ễ rộ ❦Ð♣ ❝ñ❛ dq (G) = A ❧➭ sl(2)− ♠♦❞✉❧❡ s❛♦ ❝❤♦ ♠ä✐ sl(2)− ♠♦❞✉❧❡ ❝♦♥ ❝ñ❛ A ❦❤➳❝ ❦❤➠♥❣ ✈➭ ❝ã sè ❝❤✐Ị✉ ❧í♥ ❤➡♥ ❦❤➠♥❣ s✉② ❜✐Õ♥ ✈➭ T ❧➭ sl(2)− t❤× A ♥❤❐♥ ♠ét ❞➵♥❣ s♦♥❣ t✉②Õ♥ tÝ♥❤ T ➤è✐ ①ø♥❣✱ ❜✃t ❜✐Õ♥✳ ❚❤❡♦ ❦Õt q✉➯ ị ý tr tì rộ é ủ số ▲✐❡ ❣✐❛♦ ❤♦➳♥ t♦➭♥ ♣❤➢➡♥❣ (A, T ) ψ : sl(2) −→ gl(A) ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ❜ë✐ sl(2) q✉❛ ♣❤Ð♣ ❜✐Ĩ✉ ❞✐Ơ♥ ❑Õt ❧✉❐♥ ▲✉❐♥ ✈➝♥ ❝ã ụ í tì tò ứ ột số tí t ❝➡ ❜➯♥ ❝đ❛ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❝ã ❝❤✐Ị✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ ❚r♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ❝❤ó♥❣ t➠✐ ➤➲ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ✈✃♥ ➤Ị s❛✉✿ ✶✳ ❚r×♥❤ ❜➭② ❧➵✐ ♠ét sè ❦❤➳✐ ♥✐Ư♠ ✈➭ ❦Õt q✉➯ ✈Ị ❧ý t❤✉②Õt ➤➵✐ sè ▲✐❡ tỉ♥❣ q✉➳t✳ P❤➳t ❜✐Ĩ✉ ➤Þ♥❤ ♥❣❤Ü❛ ✈Ị ➤➵✐ sè ▲✐❡ ❧ị② ❧✐♥❤✱ ➤➵✐ sè ▲✐❡ ❣✐➯✐ ➤➢ỵ❝ ✈➭ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥❀ t✐➟✉ ❝❤✉➮♥ ❧ị② ❧✐♥❤ ✭➜Þ♥❤ ❧ý ✶✳✷✳✹✮✱ t✐➟✉ ❝❤✉➮♥ ❈❛rt❛♥ ➤è✐ ✈í✐ ➤➵✐ sè ▲✐❡ ❧ị② ❧✐♥❤ ✭➜Þ♥❤ ❧ý ✶✳✷✳✼✮❀ t✐➟✉ ❝❤✉➮♥ ❣✐➯✐ ợ ị ý t rt ố số ợ ị ý t rt ➤è✐ ✈í✐ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ ✭➜Þ♥❤ ❧ý ✶✳✹✳✸✮ ✈➭ ➜Þ♥❤ ❧ý ❍✳ ❲❡②❧✱ ➜Þ♥❤ ❧ý ❈❛rt❛♥ ✲ ▲❡✈✐ ✲ ▼❛❧s❡✈ ➤è✐ ✈í✐ ➤➵✐ sè ▲✐❡ ♥ư❛ ➤➡♥ t➢➡♥❣ ø♥❣ tr♦♥❣ ❧✉❐♥ ✈➝♥ ❧➭ ➜Þ♥❤ ❧ý ✶✳✹✳✾✱ ✶✳✹✳✶✵✳ ✷✳ rì ị ĩ ề số t ❝❤✐Ị✉ t♦➭♥ ♣❤➢➡♥❣ ❝đ❛ ♠ét ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❧✐➟♥ q✉❛♥ ➤Õ♥ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣✳ ❑❤➳✐ ♥✐Ư♠ ✈Ị ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ♥ã ✭➜Þ♥❤ ❧ý ✷✳✷✳✷✱ ✷✳✷✳✸✱✮✱ ♠ét sè tÝ♥❤ ❝❤✃t ♥ã✐ ❧➟♥ ♠è✐ q✉❛♥ ❤Ư ❣✐÷❛ ➤➵✐ sè ▲✐❡ t♦➭♥ ị số ợ sè ▲✐❡ ➤➬② ➤đ ✭➜Þ♥❤ ❧ý ✷✳✷✳✹✱ ✷✳✷✳✺✮✳ ✸✳ P❤➬♥ ❝✉è✐ ❝đ❛ ❧✉❐♥ ✈➝♥ ❝❤ó♥❣ t➠✐ ①Ðt ♠ét sè ❧í♣ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❝ã ❝❤✐Ò✉ t♦➭♥ ♣❤➢➡♥❣ ❜➺♥❣ 2, ❝ơ t❤Ĩ ➤ã ❧➭ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❧ị② ❧✐♥❤ ✈➭ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ➤➬② ➤đ✳ ết q ó ợ tể ệ tr ị ❧ý ♥❤➢ ➜Þ♥❤ ❧ý ✷✳✸✳✷✱ ✷✳✸✳✸✱ ✷✳✸✳✹✱ ✷✳✸✳✾ ✈➭ ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳ ✺✵ ❚➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ế ệt ỗ ọ ệ ý tết ♥❤ã♠ ▲✐❡✱ ❇➭✐ ❣✐➯♥❣ ❙❛✉ ➤➵✐ ❤ä❝✱ ❱✐Ö♥ ❚♦➳♥ ❤ä❝ ❱✐Öt ◆❛♠✳ ❬✷❪✳ ❈❛♦ ❚r➬♥ ❚ø ❍➯✐✱ ❉➢➡♥❣ ▼✐♥❤ ❚❤➭♥❤ ✭✷✵✶✺✮✱ ❙è ❇❡tt✐ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ❝➳❝ ➤➵♦ ❤➭♠ ♣❤➯♥ ①ø♥❣ ❝đ❛ ❝➳❝ ➤➵✐ sè ▲✐❡ t♦➭♥ ♣❤➢➡♥❣ ❣✐➯✐ ➤➢ỵ❝ ❝ã sè ❝❤✐Ò✉ ≤ ❚➵♣ ❝❤Ý ❦❤♦❛ ❤ä❝ ➜❍❙P ❚P❍❈▼✱ ❙è ✺✭✼✵✮✱ tr✽✻ ✲ tr✾✻✳ ✷✳ ❚✐Õ♥❣ ❆♥❤ ❬✸❪✳ ■❣♥❛❝✐♦ ❇❛❥♦✱ ❙❛✐❞ ❇❡♥❛②❛❞✐ ✭✷✵✵✼✮✱ ▲✐❡ ❆❧❣❡❜r❛s ✇✐t❤ q✉❛❞r❛❝t✐❝ ❞✐♠❡♥s✐♦♥ ❡q✉❛❧ t♦ ✷✱ ❏♦✉r♥❛❧ ♦❢ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ❆❧❣❡❜❛r❛✱ ✷✵✾✱ ✼✷✺ ✲ ✼✸✼✳ ❬✹❪✳ ▼✳ ❇r♦❞❡♠❛♥♥ ✭✶✾✾✼✮✱ ◆♦♥❞❡❣❡♥❡r❛t❡ ✐✈❛r✐❛♥t ❜✐❧✐♥❡❛r ❢♦r♠s ♦♥ ♥♦♥❛ss♦❝✐❛✲ t✐✈❡ ❛❧❣❡❜r❛s✱ ❆❝t❛✳ ▼❛t❤✳ ❯♥✐✳ ❈♦♠❡♥✐❛♥❛❝✱ ❳▲❱■✭✷✮✱ ✶✺✶ ✲ ✷✵✶✳ ❬✺❪✳ ❏✳ ▼✳ ❋✐❣✉❡r♦❛ ❛♥❞ ❙✳ ❙t❛♥❝✐✉ ✭✶✾✾✻✮✱ ❖♥ t❤❡ str✉❝t✉r❡ ♦❢ s②♠♣❧❡❝t✐❝ s❡❧t❞✉❛❧ ▲✐❡ ❛❧❣❡❜r❛s✱ ❏✳ ▼❛t❤✳ P❤②s✱ ✸✼✭✽✮✱ ✹✶✷✶ ✲ ✹✶✸✹✳ ❬✻❪✳ ❱✳ ❑❛❝ ✭✶✾✽✺✮✱ ■♥❢✐♥✐t❡ ✲ ❞✐♠❡♥s✐♦♥❛❧ ▲✐❡ ❛❧❣❡❜r❛s✱ ❈❛♠❜r✐❣❞❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ◆❡✇ ❨♦r❦✳ ❬✼❪✳ ●✳ P✐♥❝③♦♥ ❛♥❞ ❘✳ ❯s❤✐r♦❜✐r❛ ✭✷✵✵✼✮✱ ◆❡✇ ❆♣♣❧✐❝❛t✐♦♥ ♦❢ ●r❛❞❡❞ ▲✐❡ ❆❧❣❡❜r❛s t♦ ▲✐❡ ❆❧❣❡❜r❛s✱ ●❡♥❡r❛❧✐③❡❞ ▲✐❡ ❆❧❣❡❜r❛s ❛♥❞ ❈♦❤♦♠♦❧♦❣②✱ ❏✳ ▲✐❡ ❚❤❡♦r②✱ ✶✼✱ ♣♣✳ ✻✸✸ ✲ ✻✻✼✳ ❬✽❪✳ ❉✉♦♥❣ ▼✐♥❤ ❚❤❛♥❤✳ ✭✷✵✶✸✮✱ ❚✇♦ ✲ st❡♣ ♥✐❧♣♦t❡♥t q✉❞r❛t✐❝ ▲✐❡ ❆❧❣❡❜r❛s ❛♥❞ ✽ ✲ ❞✐♠❡♥s✐♦♥❛❧ ♥♦♥ ✲ ❝♦♠♠✉t❛t✐✈❡ s②♠♠❡tr✐❝ ◆♦✈✐❦♦✈ ❛❧❣❡❜r❛s✱ ▼❛t❤❡♠❛t✐❝s✱ ❱♦❧✳ ✹✶✭✷✮✱ ♣♣✳ ✶✸✺ ✲ ✶✹✽✳ ✺✶ ❱✐❡t♥❛♠ ❏♦✉r♥❛❧ ♦❢ ... [f1 + x1 + y1 , f2 + x2 + y2 ] = (π(y1 )(f2 ) − π(y2 )(f1 ) + ϕ(x1 , x2 )) + ([x1 , x2 ]A + ψ(y1 )(x2 ) − ψ(y2 )(x1 )) + [y1 , y2 ]B ✈í✐ ♠ä✐ f1 , f2 ∈ B ∗ ; x1 , x2 ∈ A; y1 , y2 ∈ B ❑❤✐ ➤ã G ➤➢ỵ❝... −x3 x2 −x1 y2 −y1 ❙✉② r❛ adX ◦ adY ❝ã ♠❛ tr❐♥ ❜✐Ĩ✉ ❞✐Ơ♥ ❧➭ −x3 y3 − x2 y2 x2 y1 x y1 x y −x y − x y x y2 MX MY = 3 1 x1 y3 x2 y3 −x2 y2 − x1 y1 B(X, Y ) = T r(adX ◦ adY ) = ? ?2( x1 y1 + x2 y2 +... t❤Ĩ✱ Bγ Bγ : G × G −→ K, Bγ (f1 + x1 + y1 , f2 + x2 + y2 ) = T (x1 , x2 ) + γ(y1 , y2 ) + f1 (y2 ) + f2 (y1 ) ✈í✐ ♠ä✐ f1 , f2 ∈ B ∗ ; x1 , x2 ∈ A; y1 , y2 ∈ B ❈❤➢➡♥❣ ✷✳ ❈❤♦ G ✸✹ ➜➵✐ sè ▲✐❡ ❝ã ❝❤✐Ò✉