The adjoint action of a Lie group on the space of loops
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.
scientific article; zbMATH DE number 469075