The Affine Group and Semi-Direct Products

Abstract

The present chapter is devoted to the study of automorphism groups of connected Lie groups. If $\widetilde {G}$ is connected and simply connected the extension Theorem 7.13 shows that the automorphism group $\mathrm {Aut}\widetilde {G}$ is isomorphic to the group of automorphisms $\mathrm {Aut}\mathfrak {g}$ of the Lie algebra $\mathfrak {g}$, which is a Lie group. In general, the automorphism group of a connected group G is isomorphic to a closed subgroup of the automorphism group $\mathrm {Aut}\widetilde {G}$ of the universal covering $\widetilde {G}$. Therefore, AutG is a Lie group as well.