Chevalley groups associated to elliptic Lie algebras
type: Item
instance of: scholarly article
title: Chevalley groups associated to elliptic Lie algebras
zbMATH Open document ID: 0927.17011
author: Tadayoshi Takebayashi
published in: Journal of Algebra
publication date: 1999-12-15T00:00:00Z
review text: Generalizations of affine Lie algebras have been an active area of research recently and the terminology is by no means standardized. The algebras here called elliptic are also known as toroidal or extended affine algebras. The class of algebras under study in this paper is defined as follows. Let \({\mathfrak g}\) be a finite-dimensional complex simple Lie algebra. Then the associated elliptic algebra \(\hat {\mathfrak g}\) is the universal central extension of the doubly affinized algebra \({\mathfrak g} \otimes k[T^{\pm 1}, S^{\pm 1}]\), for some field \(k\). The author is concerned with the Chevalley group \(G(k[T^{\pm 1}, S^{\pm 1}])\) and the elementary subgroup \(E(k[T^{\pm 1}, S^{\pm 1}])\) generated by the root system. In contrast to the affine case, the elliptic Weyl group is not a Coxeter group and so \(E(k[T^{\pm 1}, S^{\pm 1}])\) does not have a Tits system. Most of the paper is devoted to analyzing the algebraic structure of \(E(k[T^{\pm 1}, S^{\pm 1}])\) and determining the relations of the Weyl groups of the elliptic root systems \(A_l^{(1,1)}\), \(B_l^{(1,1)}\), \(C_l^{(1,1)}\) and \(D_l^{(1,1)}\).
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 1251406
zbMATH Keywords: Chevalley groups
zbMATH Keywords: elliptic Lie algebras
zbMATH Keywords: toroidal Lie algebras
zbMATH Keywords: extended affine Lie algebras
zbMATH Keywords: Tits systems
reviewed by: Duncan J. Melville
MaRDI profile type: Publication
full work available at URL: jabr.1998.7556
OpenAlex ID: W2088809142
Wikidata QID: Q115395862
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DOI: 10.1006/JABR.1998.7556
description: scientific article; zbMATH DE number 1251406