"Kostant theorem on Lie algebra cohomology of nilpotent subalgebra"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
− | * one can use the BGG resolution and the fact that for Verma modules $H^i(\mathfrak{g}, | + | * one can use the BGG resolution and the fact that for Verma modules $H^i(\mathfrak{g},M(\mu))$ 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) | ;thm (Kostant) | ||
− | For a finite dimensional highest weight representation $ | + | Let $\lambda\in \Lambda^{+}$. For a finite dimensional highest weight representation $L({\lambda})$ of a complex semi-simple Lie algebra $\mathfrak{g}$ |
$$ | $$ | ||
− | H^k(\mathfrak{n}^{ | + | H^k(\mathfrak{n}^{-},L({\lambda}))=\bigoplus_{w\in W, \ell(w)=k}\mathbb{C}_{w\cdot \lambda} |
$$ | $$ | ||
2016년 4월 25일 (월) 23:47 판
introduction
- one can use the BGG resolution and the fact that for Verma modules $H^i(\mathfrak{g},M(\mu))$ 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)
Let $\lambda\in \Lambda^{+}$. For a finite dimensional highest weight representation $L({\lambda})$ of a complex semi-simple Lie algebra $\mathfrak{g}$ $$ H^k(\mathfrak{n}^{-},L({\lambda}))=\bigoplus_{w\in W, \ell(w)=k}\mathbb{C}_{w\cdot \lambda} $$