The Automorphism Group of a Lie Group

Abstract

Introduction. The group A (G) of all continuous and open automorphisms of a locally compact topological group G may be regarded as a topological group, the topology being defined in the usual fashion from the compact and the open subsets of G (see ?1). In general, this topological structure of A (G) is somewhat pathological. For instance, if G is the discretely topologized additive group of an infinite-dimensional vector space over an arbitrary field, then A (G) already fails to be locally compact. On the other hand, if G is a connected Lie group, we shall show without any difficulty that the compact-open topology of A (G) coincides with the topology obtained by identifying A (G) with a closed subgroup of the linear group of automorphisms of the Lie algebra of G, as was done by Chevalley (in [1]) in order to make A (G) into a Lie group. We shall then deduce that A (G) is a Lie group whenever the group of its components, G/Go, is finitely generated('), where Go denotes the component of the identity element in G. The other questions with which we shall be concerned are the following: Let I(Go) denote the group of the inner automorphisms of Go, and let E(Go, G) denote the natural image in A (Go) of A (G). Regard I(Go) and E(Go, G) as