The real cohomology of compact disconnected Lie groups

Abstract:

Let G be a compact Lie group with identity component ${G_0}$ and component group $\Gamma = G/{G_0}$. The homomorphism $\chi :G \to {\operatorname {Aut(}}{G_0})$ defined by $\chi (g)(x) = {g^{ - 1}}xg$ induces $\chi :\Gamma \to {\operatorname {Aut}}(G)/{\operatorname {Int}}(G)$. The problem of computing the real cohomology ${H^\ast }(G)$ is solved in the sense that, given $\chi$, the decomposition of $\mathfrak {G}$—the Lie algebra of ${G_0}$, and a description of $d\chi {(\gamma )_e} \in {\operatorname {Aut}}(\mathfrak {G})$, for each $\gamma \in \Gamma$, with respect to that decomposition, one can write down a complete description of ${H^\ast }(G)$ as a Hopf algebra.