On the cohomology of compact Lie groups
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.
scientific article; zbMATH DE number 871681
scientific article