Fixed points of automorphisms of compact Lie groups

Abstract

Hopf's proof that the real Cech cohomology H*(G) of a compact, connected Lie group G is an exterior algebra with odd-dimensional generators was followed by a demonstration that the number of such generators is equal to the rank of the group, that is, to the dimension of a maximal torus. We show that the latter result is a special case of a relationship between an automorphism of such a group and the automorphism it induces on the cohomology. 1* Introduction* For a set X and a function f:X—> X, let Φ(f) denote the set of fixed points of /: those x e X for which f(x) = x. If X is a topological group and / is a homomorphism, we use the symbol Φ0(f) for the component of the group Φ(f) which contains the identity element of X. By a graded vector space V we mean a sequence {VQ, Vlf V2, •••} of (real) vector spaces. The dimension of V is the sum of the dimensions of the F*. A subspace of V is a graded vector space V = {V[} such that F is a subspace of Vz; for all i. Now let G be a compact, connected Lie group and let h be an automorphism of G. Denote by PH*(G) the graded vector subspace of primitives 1 in the Hopf algebra H*(G) and let Ph* be the restriction to PH*(G) of the automorphism h*: H*(G)-> H*(G) induced by h. The main result of this paper is