A finiteness theorem for affine Lie algebras
type: Item
instance of: scholarly article
title: A finiteness theorem for affine Lie algebras
zbMATH Open document ID: 0616.17009
DOI: 10.1016/0021-8693(86)90143-2
author: Vinay V. Deodhar
author: S. Kumaresan
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let \(G(A)\) be the Kac-Moody Lie algebra associated to a symmetrizable generalized Cartan matrix A, H be a Cartan subalgebra, \(\Phi\) be the root system with basis \(\Pi\), \(\Gamma =\sum_{\alpha \in \Phi}{\mathbb{Z}}\alpha\) be the root lattice, (.,.) be the bilinear form and \(\rho \in H^*\) be such that \(2(\rho,\alpha) = (\alpha,\alpha)\) for all \(\alpha\in \Pi\). For \(\lambda \in H^*\) define the following important set: \[ X_{\lambda} = \{\mu\in \lambda +\Gamma; \quad (\mu+\rho,\mu+\rho) = (\lambda+\rho,\lambda+\rho)\}. \] By using the fact that the Casimir operator \(\Omega\) acts as a scalar \((\lambda+\rho, \lambda+\rho)\) on a highest weight module M(\(\lambda)\), Kac proved that the highest weights of all irreducible subquotients of M(\(\lambda)\) belong to \(X_{\lambda}\). The action of \(\Omega\) is no longer a scalar on a module M in the category \({\mathcal O}\) of Bernstein-Gelfand-Gelfand, instead one has the decomposition \(M= \oplus_{c\in C}M_ c\) where \(M_ c=\{m\in M\); \((\Omega -cI)^ r_ m=0\) for some \(r\}\). In order to decompose \(M_ c\), an equivalence \(\sim\) was introduced on \(H^*\) in [\textit{V. V. Deodhar}, \textit{O. Gabber} and \textit{V. Kac}, Adv. Math. 45, 92-116 (1982; Zbl 0491.17008)] and one gets \(M_ c=\oplus_{\Theta \subset X_{\lambda}}M_{\Theta}\) if \(c=(\lambda +\rho, \lambda +\rho)\), where \(M_{\Theta}\) is the submodule associated to an equivalence class \(\Theta\). The main result of this article is to prove that in the case of an affine Kac-Moody Lie algebra, \(X_{\lambda}\) contains only finitely many equivalence classes (up to the action of the Weyl group W). The authors announce that the hyperbolic case has been settled by Deodhar and Moody. Let us look at some technical parts of the paper. A complicated condition called (*) was introduced in Deodhar-Gabber-Kac's paper and they proved that \(\{\lambda,\mu\}\) satisfies (*) iff the irreducible module \(L(\mu)\) occurs as a subquotient of \(M(\lambda)\); then …
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 3998053
zbMATH Keywords: Weyl group
zbMATH Keywords: Casimir operator
zbMATH Keywords: highest weight module
zbMATH Keywords: affine Kac-Moody Lie algebra
zbMATH Keywords: equivalence classes
MaRDI profile type: Publication
Wikidata QID: Q115366824
full work available at URL: 0021-8693(86)90143-2
OpenAlex ID: W2006106738
cites work: Q5580319
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description: scientific article; zbMATH DE number 3998053