On the cohomology of compact Lie groups
type: Item
instance of: scholarly article
title: On the cohomology of compact Lie groups
zbMATH Open document ID: 0865.57040
author: Mark Reeder
published in: L'Enseignement Mathématique. 2e Série
publication date: 1997-07-13T00:00:00Z
review text: This well-written expository paper presents a new proof of the well-known results about the real cohomology ring of a compact Lie group, avoiding the usual approaches via spectral sequences and Hopf algebras. Instead, a result of \textit{L. Solomon} [Nagoya Math. J. 22, 57-64 (1963; Zbl 0117.27104)] on differential forms invariant under finite reflection groups is used as the key ingredient. The starting point of the proof is the following observation. Let \(M\) be the set of pairs \((g,T_1)\), where \(T_1\), is a maximal torus of the given group \(G\) and \(g\in T_1\); then conjugation induces an isomorphism of cohomology rings \(H(M)\cong H(G)\). Solomon's result is applied in computing \(H(M)\) as the ring of \(W\)-invariants in \(H(G/T)\otimes H(T)\), where \(T\) is a fixed maximal torus and \(W\) is the Weyl group. Here, \(H(G/T)\) is the space of \(W\)-harmonic polynomials on the Lie algebra of \(T\); this theorem of Borel is proved using Morse theory.
Mathematics Subject Classification ID: 57T10
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 57T15
zbMATH DE Number: 871681
zbMATH Keywords: invariant theory
zbMATH Keywords: real cohomology ring
zbMATH Keywords: compact Lie group
zbMATH Keywords: Lie algebra
reviewed by: Rainer Löwen
MaRDI profile type: Publication
Recommended article: On the cohomology of loop spaces of compact Lie groups
Recommended article: Q3317757
Recommended article: Cohomotopy of Lie groups
Recommended article: Q3331941
Recommended article: Nonabelian cohomology of compact Lie groups
Recommended article: Q4936152
Recommended article: Q3798823
Recommended article: On the cohomology of finite groups of Lie type
Recommended article: Cohomology of compactifications of algebraic groups
Recommended article: Q3844489
description: scientific article; zbMATH DE number 871681