The Automorphism Group of a Lie Group

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 [l]) 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 /(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 /(Go) and E(G0, G) as subgroups of A (Go). Are these subgroups closed in ^4(G0)? Is E(G0, G) topologically, as well as group-theoretically, isomorphic with the corresponding factor group of A (G) ? We shall show, under the assumption that G/Go is finitely generated, that these questions are related as follows: The natural continuous homomorphism of A (G) onto E(Go, G) is open if and only if E(G0, G) is closed in A (G0). Under the stronger assumption that G /Go is finite, a sufficient condition for E(G0, G) to be closed in A (Go) is that 7(G0) be closed in ^4(G0). Finally, in order to throw some light on the difficulties which are involved here, we shall give a simple example in which /(Go) and E(G0, G) are not closed in ^4(G0). In this example, G has only two components and Go is homeomorphic with Euclidean 5-space. 1. Topological preparation. We shall describe the topology of a group G in terms of a fundamental system 33 of neighborhoods V of the identity element. A system 33 of subsets of G will define a Hausdorff topology consistent with the group operations if and only if it satisfies the following conditions(2) :