Exponentials of certain completions of the unitary form of a Kac-Moody algebra
type: Item
instance of: scholarly article
title: Exponentials of certain completions of the unitary form of a Kac-Moody algebra
zbMATH Open document ID: 0658.17017
author: Kiyokazu Suto
published in: Proceedings of the Japan Academy. Series A
publication date: 1988-01-01T00:00:00Z
review text: Let \({\mathfrak g}_ R\) be a real Kac-Moody algebra corresponding to a symmetrizable generalized Cartan matrix, \({\mathfrak g}\) be its complexification. Let \({\mathfrak k}\) be the unitary form of \({\mathfrak g}\) given by the author previously [Proc. Japan Acad., Ser. A 62, 392-395 (1986; Zbl 0617.22012)]. In Proc. Japan Acad., Ser. A 63, 225-228 (1987; Zbl 0625.17011) the author defined the spaces \(H_ m(\pi)\) of the \(C^ m\)-vectors \((m=0,1,2,...,\infty,\omega)\) for two kinds of representations (\(\pi\),V) of \({\mathfrak g}\), namely the adjoint representation (ad,\({\mathfrak g})\) and irreducible highest weight representation \((\pi_{\Lambda},L(\Lambda))\). Then he showed that the action of the analytic completion \({\mathfrak k}_{\omega}\) of \({\mathfrak k}\) can be exponentiated to exp \(\pi\) (x) (x\(\in {\mathfrak k}_{\omega})\), which leaves each \(H_ m(\pi)\) invariant. In this paper the author extends the above argument to a more precise form. Let \({\mathfrak g}_ m=H_ m(ad)\) and \({\mathfrak k}_ m\) be the closure of \({\mathfrak k}\) in \({\mathfrak g}_ m\). Then, by the generating theorem of semigroups, the action \(\pi\) (x) of \({\mathfrak k}_{m+2}\) can be exponentiated to an endomorphism exp \(\pi\) (x) on \(H_ m(\pi)\). Some properties of the exponential map are also discussed. All theorems and propositions are given without proofs.
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B15
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 58B25
zbMATH DE Number: 4075351
zbMATH Keywords: space of \(C^ m\)-vectors
zbMATH Keywords: invariance
zbMATH Keywords: Kac-Moody algebra
zbMATH Keywords: unitary form
zbMATH Keywords: adjoint representation
zbMATH Keywords: exponential map
reviewed by: Hiromichi Yamada
MaRDI profile type: Publication
full work available at URL: pjaa.64.208
OpenAlex ID: W2080853869
cites work: Unitary structure in representations of infinite-dimensional groups and a convexity theorem
cites work: Groups associated with compact type subalgebras of Kac-Moody algebras
cites work: Differentiable vectors and analytic vectors in completions of certain representation spaces of a Kac-Moody algebra
DOI: 10.3792/PJAA.64.208
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description: scientific article; zbMATH DE number 4075351