"Kostant theorem on Lie algebra cohomology of nilpotent subalgebra"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
imported>Pythagoras0 |
imported>Pythagoras0 |
||
| 2번째 줄: | 2번째 줄: | ||
* one can use the BGG resolution and the fact that for Verma modules $H^i(\mathfrak{g},V(\mu))$ is $\mathbb{C}_{\mu}$ for $i=0$ for $i>0$. | * one can use the BGG resolution and the fact that for Verma modules $H^i(\mathfrak{g},V(\mu))$ is $\mathbb{C}_{\mu}$ for $i=0$ for $i>0$. | ||
* this requires knowing the BGG resolution, which is a stronger result since it carries information about homomorphisms between Verma modules | * this requires knowing the BGG resolution, which is a stronger result since it carries information about homomorphisms between Verma modules | ||
| + | ;thm (Kostant) | ||
| + | For a finite dimensional highest weight representation $V^{\lambda}$ of a complex semi-simple Lie algebra $\mathfrak{g}$ | ||
| + | $$ | ||
| + | H^i(\mathfrak{g},V^{\lambda})=\bigoplus_{w\in W, \ell(w)=k}\mathbb{C}_{w\cdot \lambda} | ||
| + | $$ | ||
| + | |||
| + | ==related items== | ||
| + | * [[BGG resolution]] | ||
| + | * [[Bott-Borel-Weil Theorem]] | ||
| + | |||
2016년 4월 18일 (월) 18:08 판
introduction
- one can use the BGG resolution and the fact that for Verma modules $H^i(\mathfrak{g},V(\mu))$ is $\mathbb{C}_{\mu}$ for $i=0$ for $i>0$.
- this requires knowing the BGG resolution, which is a stronger result since it carries information about homomorphisms between Verma modules
- thm (Kostant)
For a finite dimensional highest weight representation $V^{\lambda}$ of a complex semi-simple Lie algebra $\mathfrak{g}$ $$ H^i(\mathfrak{g},V^{\lambda})=\bigoplus_{w\in W, \ell(w)=k}\mathbb{C}_{w\cdot \lambda} $$