Exponentials of certain completions of the unitary form of a Kac-Moody algebra and associated groups
type: Item
instance of: scholarly article
title: Exponentials of certain completions of the unitary form of a Kac-Moody algebra and associated groups
zbMATH Open document ID: 0716.17024
author: Kiyokazu Suto
published in: Journal of Mathematics of Kyoto University
publication date: 1990-01-01T00:00:00Z
review text: In the previous paper [J. Math. Kyoto Univ. 28, 633-659 (1988; Zbl 0683.17013)] the author considered certain classes of representations (\(\pi\),V) of a complex Kac-Moody algebra \({\mathfrak g}\) with a symmetrizable generalized Cartan matrix, and defined the spaces \(H_ m(\pi)\) of \(C^ m\)-vectors \((m=0,1,2,...,\infty,\omega)\) in completions of V. Let \({\mathfrak k}\) be the unitary form of \({\mathfrak g}\). Denote by \({\mathfrak k}_ m\) the closure of \({\mathfrak k}\) in the space \(H_ m(ad)\) of \(C^ m\)-vectors for the adjoint representation. In the paper under review the author proves the exponentiability of the elements in \({\mathfrak k}_{m+2}\) acting on \(H_ m(\pi)\) (Theorem 4.8), and the continuity of the exponential map (Theorem 5.2). For this purpose the author introduces the negative spaces \(H_{-m}(\pi)\) as the dual of \(H_ m(\pi)\) and defines the dual action \({\mathfrak k}_{m+1}\times H_{-m}(\pi)\to H_{-m-1}(\pi)\). For the action of \({\mathfrak k}_ m\) on the positive and negative spaces, he gives sharp estimates for the norm as linear maps \(H_ m(\pi)\to H_{m- 1}(\pi)\) and \(H_{-m+2}(\pi)\to H_{-m+1}(\pi)\). These estimates ensure that one can apply the criterion for exponentiability of an operator, which appears in Yosida's book on functional analysis.
reviewed by: Hiromichi Yamada
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B15
zbMATH DE Number: 4179614
zbMATH Keywords: representations
zbMATH Keywords: complex Kac-Moody algebra
zbMATH Keywords: exponential map
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250520123
OpenAlex ID: W1541417612
DOI: 10.1215/KJM/1250520123
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description: scientific article; zbMATH DE number 4179614