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| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
| − | * | + | * <math>L(\cdot)</math> : simple module, <math>V(\cdot)</math> : Verma module |
| − | * Weyl character formula. For | + | * Weyl character formula. For <math>\lambda\in \Lambda^{+}</math>, |
| − | + | :<math> | |
\operatorname{ch}L(\lambda)=\sum_{w \in W}(-1)^{\ell(w)}\operatorname{ch} M(w\cdot \lambda) | \operatorname{ch}L(\lambda)=\sum_{w \in W}(-1)^{\ell(w)}\operatorname{ch} M(w\cdot \lambda) | ||
| − | + | </math> | |
* goal : realize this formula as an Euler characteristic | * goal : realize this formula as an Euler characteristic | ||
| − | * The BGG resolution resolves a finite-dimensional simple | + | * The BGG resolution resolves a finite-dimensional simple <math>\mathfrak{g}</math>-module <math>L(\lambda)</math> by direct sums of Verma modules indexed by weights "of the same length" in the orbit <math>W\cdot \lambda</math> |
;thm (Bernstein-Gelfand-Gelfand Resolution). | ;thm (Bernstein-Gelfand-Gelfand Resolution). | ||
| − | Let | + | Let <math>\lambda\in \Lambda^{+}</math>. There is an exact sequence of Verma modules |
| − | + | :<math> | |
| − | 0 \to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots M({\lambda})\to L({\lambda})\to 0 | + | 0 \to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots \to M({\lambda})\to L({\lambda})\to 0 |
| − | + | </math> | |
| − | where | + | where <math>\ell(w)</math> is the length of the Weyl group element <math>w</math>, <math>w_0</math> is the Weyl group element |
| − | of maximal length. Here | + | of maximal length. Here <math>\rho</math> is half the sum of the positive roots. |
==applications== | ==applications== | ||
| − | * This is used to compute the cohomologies of | + | * This is used to compute the cohomologies of <math>\mathfrak{n}^+</math>. |
* see [[Kostant theorem on Lie algebra cohomology of nilpotent subalgebra]] | * see [[Kostant theorem on Lie algebra cohomology of nilpotent subalgebra]] | ||
| 46번째 줄: | 46번째 줄: | ||
==articles== | ==articles== | ||
| + | * Pierre Julg, The Bernstein-Gelfand-Gelfand complex for rank one semi simple Lie groups as a Kasparov module, arXiv:1605.07408 [math.OA], May 24 2016, http://arxiv.org/abs/1605.07408 | ||
* Griffeth, Stephen, and Emily Norton. “Character Formulas and Bernstein-Gelfand-Gelfand Resolutions for Cherednik Algebra Modules.” arXiv:1511.00748 [math], November 2, 2015. http://arxiv.org/abs/1511.00748. | * Griffeth, Stephen, and Emily Norton. “Character Formulas and Bernstein-Gelfand-Gelfand Resolutions for Cherednik Algebra Modules.” arXiv:1511.00748 [math], November 2, 2015. http://arxiv.org/abs/1511.00748. | ||
* Zelevinski, Resolvents, dual pairs, and character formulas http://www.ms.unimelb.edu.au/~ram/Resources/ResolventsDualPairsAndCharacterFormulas.html | * Zelevinski, Resolvents, dual pairs, and character formulas http://www.ms.unimelb.edu.au/~ram/Resources/ResolventsDualPairsAndCharacterFormulas.html | ||
| − | * [34] A. Rocha-Caridi, Splitting Criteria for | + | * [34] A. Rocha-Caridi, Splitting Criteria for <math>\mathfrak{g}</math>-modules induced from a parabolic and the Bernstein-Gelfand-Gelfand resolution of a finite-dimensional, irreducible <math>\mathfrak{g}</math>-module, Trans. Amer. Math. Soc.262 (1980), no. 2, 335–366 |
* [26] G. Kempf, The Grothendieck-Cousin complex of an induced representation , Advances in Mathematics 29 (1978), 310–396 | * [26] G. Kempf, The Grothendieck-Cousin complex of an induced representation , Advances in Mathematics 29 (1978), 310–396 | ||
* [31] Lepowsky, J. “A Generalization of the Bernstein-Gelfand-Gelfand Resolution.” Journal of Algebra 49, no. 2 (1977): 496–511. | * [31] Lepowsky, J. “A Generalization of the Bernstein-Gelfand-Gelfand Resolution.” Journal of Algebra 49, no. 2 (1977): 496–511. | ||
* J. Bernstein, I. Gel'fand, and S. Gel'fand, A category of g-modules, Functional Anal. Appl. 10 (1976), 87-92 | * J. Bernstein, I. Gel'fand, and S. Gel'fand, A category of g-modules, Functional Anal. Appl. 10 (1976), 87-92 | ||
| − | * [5] Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Differential Operators on the Base Affine Space and a Study of | + | * [5] Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Differential Operators on the Base Affine Space and a Study of <math>\mathfrak{g}</math>-Modules’. In Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), 21–64. Halsted, New York, 1975. http://www.ams.org/mathscinet-getitem?mr=0578996. |
* Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Structure of Representations That Are Generated by Vectors of Highest Weight’. Akademija Nauk SSSR. Funkcional\cprime Nyi Analiz I Ego Priloženija 5, no. 1 (1971): 1–9. | * Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Structure of Representations That Are Generated by Vectors of Highest Weight’. Akademija Nauk SSSR. Funkcional\cprime Nyi Analiz I Ego Priloženija 5, no. 1 (1971): 1–9. | ||
[[분류:Lie theory]] | [[분류:Lie theory]] | ||
[[분류:abstract concepts]] | [[분류:abstract concepts]] | ||
| + | [[분류:migrate]] | ||
2020년 11월 16일 (월) 05:25 기준 최신판
introduction
- \(L(\cdot)\) : simple module, \(V(\cdot)\) : Verma module
- Weyl character formula. For \(\lambda\in \Lambda^{+}\),
\[ \operatorname{ch}L(\lambda)=\sum_{w \in W}(-1)^{\ell(w)}\operatorname{ch} M(w\cdot \lambda) \]
- goal : realize this formula as an Euler characteristic
- The BGG resolution resolves a finite-dimensional simple \(\mathfrak{g}\)-module \(L(\lambda)\) by direct sums of Verma modules indexed by weights "of the same length" in the orbit \(W\cdot \lambda\)
- thm (Bernstein-Gelfand-Gelfand Resolution).
Let \(\lambda\in \Lambda^{+}\). There is an exact sequence of Verma modules \[ 0 \to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots \to M({\lambda})\to L({\lambda})\to 0 \] where \(\ell(w)\) is the length of the Weyl group element \(w\), \(w_0\) is the Weyl group element of maximal length. Here \(\rho\) is half the sum of the positive roots.
applications
- This is used to compute the cohomologies of \(\mathfrak{n}^+\).
- see Kostant theorem on Lie algebra cohomology of nilpotent subalgebra
generalization of BGG resolution
- There exist generalizations to symmetrizable Kac-Moody algebras, cf. [34].
- Kempf obtained a resolution of finite-dimensional L(λ) in terms of the Grothendieck-Cousin complex in [26], which is dual to the BGG resolution.
- This was extended by Kumar to arbitrary Kac-Moody algebras; he thus obtained the BGG resolution here, and computed the Weyl-Kac character formula and the cohomologies of n+ (cf. [30,§9.3]).
- Talk on BGG resolution
- Verma modules
- BGG reciprocity
- BGG category
- Kostant theorem on Lie algebra cohomology of nilpotent radical
- Bott-Borel-Weil Theorem
- Koszul complex
books
- [30] Shrawan Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory, Birkhauser, Progress in Math. 204, Boston, 2002
- James E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Grad. Stud. Math., 94, Amer. Math. Soc., Providence, RI, 2008.
expositions
- http://rvirk.com/notes/student/catObasics.pdf
- BGG resolution http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf
- Wang, Jing Ping. “Representations of sl(2,C) in the BGG Category O and Master Symmetries.” arXiv:1408.3437 [nlin], August 14, 2014. http://arxiv.org/abs/1408.3437.
- http://stanford.edu/~khare/EoM-BGG-O.pdf
articles
- Pierre Julg, The Bernstein-Gelfand-Gelfand complex for rank one semi simple Lie groups as a Kasparov module, arXiv:1605.07408 [math.OA], May 24 2016, http://arxiv.org/abs/1605.07408
- Griffeth, Stephen, and Emily Norton. “Character Formulas and Bernstein-Gelfand-Gelfand Resolutions for Cherednik Algebra Modules.” arXiv:1511.00748 [math], November 2, 2015. http://arxiv.org/abs/1511.00748.
- Zelevinski, Resolvents, dual pairs, and character formulas http://www.ms.unimelb.edu.au/~ram/Resources/ResolventsDualPairsAndCharacterFormulas.html
- [34] A. Rocha-Caridi, Splitting Criteria for \(\mathfrak{g}\)-modules induced from a parabolic and the Bernstein-Gelfand-Gelfand resolution of a finite-dimensional, irreducible \(\mathfrak{g}\)-module, Trans. Amer. Math. Soc.262 (1980), no. 2, 335–366
- [26] G. Kempf, The Grothendieck-Cousin complex of an induced representation , Advances in Mathematics 29 (1978), 310–396
- [31] Lepowsky, J. “A Generalization of the Bernstein-Gelfand-Gelfand Resolution.” Journal of Algebra 49, no. 2 (1977): 496–511.
- J. Bernstein, I. Gel'fand, and S. Gel'fand, A category of g-modules, Functional Anal. Appl. 10 (1976), 87-92
- [5] Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Differential Operators on the Base Affine Space and a Study of \(\mathfrak{g}\)-Modules’. In Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), 21–64. Halsted, New York, 1975. http://www.ams.org/mathscinet-getitem?mr=0578996.
- Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Structure of Representations That Are Generated by Vectors of Highest Weight’. Akademija Nauk SSSR. Funkcional\cprime Nyi Analiz I Ego Priloženija 5, no. 1 (1971): 1–9.