Extended affine root system. IV: Simply-laced elliptic Lie algebras
type: Item
instance of: scholarly article
title: Extended affine root system. IV: Simply-laced elliptic Lie algebras
zbMATH Open document ID: 0987.17012
author: Kyoji Saito
author: Daigo Yoshii
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2002-06-30T00:00:00Z
review text: For Part III, cf. ibid. 33, 301--329 (1997; Zbl 0901.20016), Part V, see CRM Proc. Lect. Notes. 30, 185--222 (2001; Zbl 1015.11013). The notion of generalized root system was introduced by the first author [Part I, Publ. Res. Inst. Math. Sci. 21, 75--179 (1985; Zbl 0573.17012)]. If the Witt index of the root lattice is 0 or 1, then the root system is either finite, affine or hyperbolic. If the Witt index is 2, the root system is called an elliptic root system. In this paper the authors study three Lie algebras associated with a simply-laced elliptic root system \(R\): (1) a certain subalgebra of the quotient Lie algebra of the lattice vertex algebra of \textit{R. Borcherds} [Proc. Natl. Acad. Sci. USA 83, 3068--3071 (1986; Zbl 0613.17012)] for the elliptic root lattice \(Q(R)\), (2) a Lie algebra presented by the Chevalley generators and the generalized Serre relations attached to the elliptic Dynkin diagram, (3) an amalgamation of an affine Heisenberg algebra and an affine Kac-Moody algebra. The main result is that these three Lie algebras are isomorphic. A good survey of the theory of generalized root systems can be found in the introduction of this paper.
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 1567204
zbMATH Keywords: affine Lie algebra
zbMATH Keywords: generalized root system
zbMATH Keywords: simply-laced elliptic root system
zbMATH Keywords: lattice vertex algebra
zbMATH Keywords: Chevalley generators
zbMATH Keywords: generalized Serre relations
zbMATH Keywords: elliptic Dynkin diagram
zbMATH Keywords: affine Mac-Moody algebra
reviewed by: Hiromichi Yamada
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195142952
OpenAlex ID: W2082091979
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DOI: 10.2977/PRIMS/1195142952
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description: scientific article; zbMATH DE number 1567204