"BGG resolution"의 두 판 사이의 차이

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-==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
-$$
-\to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots \to M({\lambda})\to L({\lambda})\to 0
-$$
-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]).
-==related items==
-* [[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.
-[[분류:Lie theory]]
-[[분류:abstract concepts]]
-[[분류:migrate]]

"BGG resolution"의 두 판 사이의 차이

2020년 11월 12일 (목) 03:52 판

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