The adjoint action of a Lie group on the space of loops
type: Item
instance of: scholarly article
title: The adjoint action of a Lie group on the space of loops
zbMATH Open document ID: 0795.57015
author: Akira Kono
author: Kazumoto Kozima
published in: Journal of the Mathematical Society of Japan
publication date: 1993-12-20T00:00:00Z
review text: Let \(G\) be a compact simply-connected Lie group, and \(p\) a prime. The main theorem is that the following three conditions are equivalent. (1) \(H^*(G;\mathbb{Z})\) is \(p\)-torsion free; (2) The adjoint map \(\text{Ad} : G \times \Omega G \to \Omega G\) induces the same morphism of \(\bmod p\) homology groups as does the projection onto the second factor; (3) \(H^*(B\Lambda G;\mathbb{Z}/p)\) is isomorphic as an algebra to \(H^*(BG; \mathbb{Z}/p) \otimes H^*(G;\mathbb{Z}/p)\). Here Ad is defined by \(\text{Ad} (g,l) (t) = gl(t)g^{-1}\), and \(\Lambda G\) denotes the space of free loops on \(G\). The proof of the equivalence of (1) and (2) uses the classification of finite simple Lie algebras.
Mathematics Subject Classification ID: 57T10
zbMATH DE Number: 469075
zbMATH Keywords: mod \(p\) homology
zbMATH Keywords: Lie group
zbMATH Keywords: adjoint map
zbMATH Keywords: space of free loops
reviewed by: Donald M. Davis
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04530495
OpenAlex ID: W2067585051
Wikidata QID: Q115225516
DOI: 10.2969/JMSJ/04530495
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description: scientific article; zbMATH DE number 469075