"Kostant theorem on Lie algebra cohomology of nilpotent subalgebra"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| (다른 사용자 한 명의 중간 판 2개는 보이지 않습니다) | |||
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==introduction== | ==introduction== | ||
* Humphreys 6.6 | * 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 | + | * 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 <math>W</math> 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. | * 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 | + | * 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 <math>\mathfrak{g}</math> (conjugate under the adjoint group to <math>\mathfrak{n}</math> or <math>\mathfrak{n}^-</math>) with coefficients in a finite dimensional simple module <math>L(\lambda)</math>. |
* Kostant [197] later developed these ideas further in the setting of Lie algebra cohomology. | * 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 | + | * one can use the BGG resolution and the fact that for Verma modules <math>H^i(\mathfrak{g},M(\mu))</math> is <math>\mathbb{C}_{\mu}</math> for <math>i=0</math> for <math>i>0</math>. |
* 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) | ||
| − | Let | + | Let <math>\lambda\in \Lambda^{+}</math>. For a finite dimensional highest weight representation <math>L({\lambda})</math> of a complex semi-simple Lie algebra <math>\mathfrak{g}</math> |
| − | + | :<math> | |
H^k(\mathfrak{n}^{-},L({\lambda}))=\bigoplus_{w\in W, \ell(w)=k}\mathbb{C}_{w\cdot \lambda} | H^k(\mathfrak{n}^{-},L({\lambda}))=\bigoplus_{w\in W, \ell(w)=k}\mathbb{C}_{w\cdot \lambda} | ||
| − | + | </math> | |
==related items== | ==related items== | ||
| 22번째 줄: | 22번째 줄: | ||
[[분류:Lie theory]] | [[분류:Lie theory]] | ||
| + | [[분류:migrate]] | ||
2020년 11월 16일 (월) 05:35 기준 최신판
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 \(\mathfrak{g}\) \[ H^k(\mathfrak{n}^{-},L({\lambda}))=\bigoplus_{w\in W, \ell(w)=k}\mathbb{C}_{w\cdot \lambda} \]