Weight spaces and root spaces of Kac-Moody algebras
type: Item
instance of: scholarly article
title: Weight spaces and root spaces of Kac-Moody algebras
zbMATH Open document ID: 0813.17018
published in: Journal of Algebra
publication date: 1994-09-28T00:00:00Z
review text: Let \({\mathfrak g} (A)\) be the Kac-Moody algebra associated to a symmetric generalized Cartan matrix. The purpose of the paper under review is to give an explicit realization of the weight spaces of an integrable highest weight \({\mathfrak g} (A)\)-module \(L(\lambda)\) and the weight spaces of the universal enveloping algebra \(U({\mathfrak n}_ -)\) in terms of certain spaces of rational functions. The approach is based on an embedding of \(L(\lambda)\) in the (shifted) Fock space \(V_ \lambda\) constructed from the root lattice of \({\mathfrak g} (A)\) [see \textit{R. E. Borcherds}, Proc. Natl. Acad. Sci. USA 84, 3068-3071 (1986; Zbl 0613.17012)]. It is difficult to characterize the weight subspace \(L(\lambda)_{\lambda - \ell_ 1 \alpha_ 1 - \cdots - \ell_ N \alpha_ N}\) within \(V_ \lambda\), and the main results are obtained by passing to the matrix coefficients \(x \mapsto \langle e^ \lambda\), \(\prod Y(e^{\alpha_ j}\), \(z_{j,i})x \rangle\), \(1 \leq j \leq N\), \(l \leq i \leq \ell_ j\), for vertex operators, and characterizing the weight subspace in terms of this map.
reviewed by: Mirko Primc
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B69
zbMATH DE Number: 645115
zbMATH Keywords: root spaces
zbMATH Keywords: integrable highest weight module
zbMATH Keywords: Kac-Moody algebra
zbMATH Keywords: explicit realization
zbMATH Keywords: weight spaces
zbMATH Keywords: universal enveloping algebra
zbMATH Keywords: vertex operators
author: Yongchang Zhu
MaRDI profile type: Publication
full work available at URL: jabr.1994.1263
OpenAlex ID: W1966097241
DOI: 10.1006/JABR.1994.1263
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description: scientific article; zbMATH DE number 645115