On the almost algebraicity of groups of automorphisms of connected Lie groups

TL;DR

The paper addresses when the automorphism group ${ m Aut}(G)$ of a connected Lie group $G$ is almost algebraic, focusing on nontrivial centers. It proves a precise criterion: if the group of automorphism restrictions to the center’s maximal torus $C$ is finite, then ${ m Aut}(G)$ is almost algebraic exactly when either $C$ is trivial or the abelian quotient $A=G/ar{[G,G]}C$ is simply connected; equivalently, ${ m Hom}(A,C)$ is connected. Central to the analysis are spiral shear automorphisms arising from ${ m Hom}(A,C)$, and a torus-fixing criterion for subgroups fixing a torus pointwise. The results yield clean descriptions for class $ extbf{C}$ groups and provide a rich set of examples illustrating when almost algebraicity holds or fails, clarifying the obstructions and offering practical tools for studying ${ m Aut}(G)$ and its subgroups.

Abstract

Let $G$ be a connected Lie group, $C$ be the maximal compact connected subgroup of the center of $G$, and let Aut$(G)$ denote the group of Lie automorphisms of $G$, viewed, canonically, also as a subgroup of GL$(\frak G)$, where $\frak G$ is the Lie algebra of $G$. It is known (see Dani (1992) and Previts-Wu (2001)) that when $C$ is trivial Aut$(G)$ is almost algebraic, in the sense that it is open in an algebraic subgroup of 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 identify obstructions 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 $G/\overline{[G,G]}C$ (see Theorem 1.1 for details). For the class of Lie groups which admit a finite-dimensional representation with discrete kernel (called class $\mathcal{C}$ groups) this yields a more precise description as to when Aut$(G)$ is almost algebraic (see Corollary 1.3), while in the general case a variety of patterns are observed (see §6). Along the way we also study almost algebraicity of subgroups of Aut$(G)$ fixing each point of a given torus in $G$, containing $C$ (see in particular Theorem 1.5), which also turns out to be of independent interest.

Theorems & Definitions (41)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Corollary 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 2.1
• proof
• Lemma 2.2
• proof