Some automorphisms of generalized Kac-Moody algebras
type: Item
instance of: scholarly article
title: Some automorphisms of generalized Kac-Moody algebras
zbMATH Open document ID: 0885.17016
author: Jürgen Fuchs
author: Urmie Ray
author: Christoph Schweigert
published in: Journal of Algebra
publication date: 1998-04-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/q-alg/9605046
review text: Generalized Kac-Moody algebras are similar in many ways to ordinary Kac-Moody algebras, but allow for the possibility of imaginary simple roots. In this paper, the authors extend the results of \textit{J. Fuchs, B. Schellekens} and \textit{C. Schweigert} [Commun. Math. Phys. 180, No. 1, 39-97 (1996; Zbl 0863.17020)] to cover the case of symmetrizable generalized Kac-Moody algebras, and they greatly improve the theory behind some detailed results obtained by case analysis in the earlier paper. In more detail, given a generalized Kac-Moody algebra \(G\) with associated Cartan matrix \(A\) and index set \(I\), the authors consider a permutation of \(I\) fixing \(A\). This permutation induces an automorphism \(\omega\) of \(G\) preserving the usual triangular decomposition and in turn inducing a bijection on the dual of the Cartan subalgebra of \(G\). From \(\omega\), the authors obtain a linear map of \(G\)-modules, denoted \(\tau_{\omega}\), which in general is not a homomorphism, but a twining, or twisted homomorphism. However, the authors' main interest is in the situation where \(\tau_{\omega}\) is an isomorphism of (given) Verma modules and irreducible highest weight modules. In this case, they provide an explicit determination of the `twining' character in terms of the ordinary character of an associated Verma (or irreducible) module of a related Lie algebra called the orbit algebra.
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 81R10
zbMATH DE Number: 1021749
zbMATH Keywords: generalized Kac-Moody algebra
zbMATH Keywords: automorphism
zbMATH Keywords: twining character
zbMATH Keywords: orbit Lie algebra
zbMATH Keywords: Verma modules
zbMATH Keywords: highest weight module
zbMATH Keywords: conformal field theory
reviewed by: Duncan J. Melville
MaRDI profile type: Publication
OpenAlex ID: W2085115664
arXiv ID: q-alg/9605046
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DOI: 10.1006/JABR.1996.6907
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description: scientific article