On the almost algebraicity of groups of automorphisms of connected Lie groups
Let $G$ be a connected Lie group, $C$ be the maximal compact connected subgroup of the center of $G$, and let ${\rm Aut}(G)$ denote the group of Lie automorphisms of $G$, viewed, canonically, also as a subgroup of ${\rm GL} (\frak G)$, where $\frak G$ is the Lie algebra of $G$. It is known that when $C$ is trivial ${\rm Aut}(G)$ is almost algebraic, in the sense that it is of finite index in an algebraic subgroup of ${\rm GL}(\frak G)$, and in particular has only finitely many connected components. In this paper we analyse the situation further in this respect, with $C$ possibly nontrivial, and describe necessary and sufficient conditions for almost algebraicity to hold; the criteria are in terms of the group of restrictions of automorphisms of $G$ to $C$, and the abelian quotient Lie group $G/\overline{[G,G]}C$. For the class of Lie groups which admit a finite-dimensional representation with discrete kernel, a more specific criterion for ${\rm Aut}(G)$ to be almost algebraic is obtained, while in the general case a variety of patterns are illustrated through examples. Along the way we also study almost algebraicity of subgroups of ${\rm Aut}(G)$ fixing each point of a given torus in $G$, containing $C$, which also turns out to be of independent interest.