Loop groups and their actions on the corresponding completed affine Lie algebras
type: Item
instance of: scholarly article
title: Loop groups and their actions on the corresponding completed affine Lie algebras
zbMATH Open document ID: 0785.22026
author: Kiyokazu Suto
published in: Journal of Mathematics of Kyoto University
publication date: 1993-05-16T00:00:00Z
review text: The paper under review is devoted to the study of groups of \(C^ k\)- loops in Lie groups. The author follows the method in the work of \textit{H. Garland} [Publ. Math., Inst. Haut. Étud. Sci. 52, 5-136 (1980; Zbl 0475.17004)]. Let \(L_ k\) be the space \(C^ k(S^ 1)\) and \(L_{k,\mathbb{R}}\) the subspace consisting of real functions (\(k=0,1,\dots,\infty\)). For a finite-dimensional manifold \(M\), let \(M(L_ k)\) denote the loop space of class \(C^ k\). In the first part of the paper the structure of \(M(L_ k)\) is studied. The main result here is: if \(M\) is a Lie group, then \(M(L_ k)\) can be provided with a canonical Lie group structure such that its tangent space is \((L_{k,\mathbb{R}})^{\dim M}\). Now let \(G\) be a finite-dimensional connected simply connected complex simple Lie group, \({\mathfrak g}\) the Lie algebra of \(G\). Denote \(\widetilde{G}_ k= G(L_ k)\), \(\widetilde {\mathfrak g}_ k={\mathfrak g}(L_ k)\). Let \(\widehat{\mathfrak g}_ k\) be a corresponding completed affine Lie algebra -- a one-dimensional central extension of \(\widetilde{\mathfrak g}_ k\). The group \(\widetilde{G}_ k\) acts in a natural way on \(\widetilde {\mathfrak g}_ k\) (by the adjoint representation). In the second part of the paper the author extends this action of \(\widetilde{G}_ k\) to an action on \(\widehat{\mathfrak g}_ k\) (with an explicit description). Moreover, he extends the action on \(\widehat {\mathfrak g}_ k\) to that on the so called extended affine Lie algebra \(\widehat{\mathfrak g}_ k^ e\). As an application explicit forms are given of the normalizers and centralizers in \(\widetilde{G}_ k\) of the Cartan subalgebras of \(\widehat {\mathfrak g}_ k\) and \(\widehat {\mathfrak g}_ k^ e\). It turns out that in both cases the corresponding quotient groups are canonically isomorphic to the usual affine Weyl group.
Mathematics Subject Classification ID: 22E67
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 169032
zbMATH Keywords: loop groups
zbMATH Keywords: Kac-Moody algebras
zbMATH Keywords: Kac-Moody groups
zbMATH Keywords: complex simple Lie group
zbMATH Keywords: affine Lie algebra
zbMATH Keywords: normalizers
zbMATH Keywords: centralizers
zbMATH Keywords: Cartan subalgebras
zbMATH Keywords: affine Weyl group
reviewed by: V. F. Molchanov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519493
OpenAlex ID: W1505712502
Wikidata QID: Q115240414
DOI: 10.1215/KJM/1250519493
description: scientific article; zbMATH DE number 169032