Groups associated with unitary forms of Kac-Moody algebras
type: Item
instance of: scholarly article
title: Groups associated with unitary forms of Kac-Moody algebras
zbMATH Open document ID: 0651.17010
author: Kiyokazu Suto
published in: Journal of the Mathematical Society of Japan
publication date: 1988-01-01T00:00:00Z
review text: The group associated with Kac-Moody algebras and its unitary forms were constructed and studied by Kac-Peterson and others [cf. \textit{V. G. Kac}, Infinite dimensional Lie algebras (2nd edition, Cambridge University Press 1985; Zbl 0574.17010); Astérisque, No.Hors Ser. 1985, 165-208 (1985; Zbl 0625.22014)]. The group is constructed by the exponential map defined on the real root vectors and there do not exist the one-parameter subgroups corresponding to imaginary roots. Now, let \({\mathfrak k}\) be the unitary form of a complex Kac-Moody algebra \({\mathfrak g}\) with symmetrizable generalized Cartan matrix. In this paper, the author constructs the group associated to \({\mathfrak k}\) in such a way that the exponential map is defined for every element of the algebra \({\mathfrak k}.\) Namely, let \(\Lambda\) be a dominant integral element in the dual of the Cartan subalgebra of \({\mathfrak g}\), L(\(\Lambda)\) the irreducible highest weight module for \({\mathfrak g}\) with highest weight \(\Lambda\). Then, L(\(\Lambda)\) has a canonical pre-Hilbert space structure under which each element of \({\mathfrak k}\) acts on L(\(\Lambda)\) as an anti-symmetric operator. The author defines the exponential map from \({\mathfrak k}\) into the group U(\(\Lambda)\) of all unitary operators on the completion H(\(\Lambda)\) of L(\(\Lambda)\) equipped with the strong operator topology. The group associated with \({\mathfrak k}\) is defined to be the closed subgroup of U(\(\Lambda)\) generated by exp \({\mathfrak k}\). The author discusses the relation between the constructed group and the groups constructed on the lowest weight modules in the same manner and also the relation between these groups and the groups treated by Kac-Peterson.
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 58B25
zbMATH DE Number: 4061460
zbMATH Keywords: Kac-Moody groups
zbMATH Keywords: Kac-Moody algebras
zbMATH Keywords: exponential map
zbMATH Keywords: generalized Cartan matrix
zbMATH Keywords: highest weight module
reviewed by: Eiichi Abe
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04010085
OpenAlex ID: W2008163915
DOI: 10.2969/JMSJ/04010085
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description: scientific article; zbMATH DE number 4061460