"Kostant theorem on Lie algebra cohomology of nilpotent subalgebra"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
| + | * Humphreys 6.6 | ||
| + | * At the end of his influential 1957 paper on the cohomology of vector bundles on homogeneous spaces such as flag varieties of semisimple Lie groups, Bott [43, §15] obtained what he described as a “curious corollary”: an explicit formula in terms of $W$ for the dimensions of certain Lie algebra cohomology groups. | ||
| + | * He also remarked that this formula can be shown “without much trouble” to imply Weyl’s character formula. | ||
| + | * Although he worked in the setting of compact Lie groups and their complexifications, the essential point of the corollary is to describe the cohomology of a maximal nilpotent subalgebra of $\mathfrak{g}$ (conjugate under the adjoint group to $\mathfrak{n}$ or $\mathfrak{n}^-$) with coefficients in a finite dimensional simple module $L(\lambda)$. | ||
| + | * Kostant [197] later developed these ideas further in the setting of Lie algebra cohomology. | ||
* 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 $i>0$. | * 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 | ||
2016년 4월 25일 (월) 23:57 판
introduction
- Humphreys 6.6
- At the end of his influential 1957 paper on the cohomology of vector bundles on homogeneous spaces such as flag varieties of semisimple Lie groups, Bott [43, §15] obtained what he described as a “curious corollary”: an explicit formula in terms of $W$ for the dimensions of certain Lie algebra cohomology groups.
- He also remarked that this formula can be shown “without much trouble” to imply Weyl’s character formula.
- Although he worked in the setting of compact Lie groups and their complexifications, the essential point of the corollary is to describe the cohomology of a maximal nilpotent subalgebra of $\mathfrak{g}$ (conjugate under the adjoint group to $\mathfrak{n}$ or $\mathfrak{n}^-$) with coefficients in a finite dimensional simple module $L(\lambda)$.
- Kostant [197] later developed these ideas further in the setting of Lie algebra cohomology.
- 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 $\mathfrak{g}$ $$ H^k(\mathfrak{n}^{-},L({\lambda}))=\bigoplus_{w\in W, \ell(w)=k}\mathbb{C}_{w\cdot \lambda} $$