Classifying spaces of Kač-Moody groups
type: Item
instance of: scholarly article
title: Classifying spaces of Kač-Moody groups
zbMATH Open document ID: 1031.55009
author: Carles Broto
author: Nitu Kitchloo
published in: Mathematische Zeitschrift
publication date: 2002-09-25T00:00:00Z
review text: Kač-Moody groups are essentially obtained by integrating certain in general infinite dimensional Lie algebras. The representation theory of these Lie algebras is a priori quite closely related to the representation theory of finite dimensional Lie algebras, and the paper under review arose from asking whether these similarities are reflected in the homotopy theoretic properties of Kač-Moody groups. It turned out to be useful to investigate this question in a more general context. Let \(p\) be a fixed prime. Starting from the category \({\mathcal L}\) of compact Lie groups, the authors consider a new class of topological groups \(K_1({\mathcal L})\). By definition a compactly generated Hausdorff topological group \(G\) is contained in \(K_1({\mathcal L})\) if there is a finite \(G\)-\(CW\)-complex \(X\) whose isotropy groups are in \({\mathcal L}\) and for which for any finite \(p\)-group \(\pi \subset G\) the fixed point space \(X^\pi\) is \(p\)-acyclic. The definition is a variation of a corresponding notion introduced by \textit{P. H. Kropholler} and \textit{G. Mislin} in [Comment. Math. Helv. 73, 122-136 (1998; Zbl 0927.20033)], where these two authors considered discrete groups \(G\) rather than topological groups, and acyclic fixed point sets instead of \(p\)-acyclic fixed point sets. Kač-Moody groups are contained in \(K_1({\mathcal L})\), however the class \(K_1({\mathcal L})\) contains way more groups. For example it also contains all parabolic subgroups of Kač-Moody groups, and if \(G\in K_1({\mathcal L})\) then also all centralizers \(C_G(\pi)\) for finite \(p\)-subgroups \(\pi \subset G\) are in \(K_1({\mathcal L})\). The authors now show that indeed many results familiar within the context of compact Lie groups also hold or have analog versions for topological groups \(G\in K_1({\mathcal L})\). For example they show that the mod-\(p\)-cohomology algebra of \(G\) is noetherian. They also show that Lannes' \(T\)-functor computes the mod-\(p\) cohomology of…
reviewed by: Michael Joachim
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 55R37
zbMATH DE Number: 1807899
zbMATH Keywords: Kač-Moody group
zbMATH Keywords: Kač-Moody algebra
zbMATH Keywords: Lannes \(T\)-functor
zbMATH Keywords: centralizer decomposition of classifying space
MaRDI profile type: Publication
full work available at URL: s002090100391
OpenAlex ID: W2043395371
DOI: 10.1007/S002090100391
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description: scientific article; zbMATH DE number 1807899