Minimality of the inner automorphism group

TL;DR

This paper investigates the conditions under which the inner automorphism group of a connected Lie group is minimal, introducing the $z$-Minimality Criterion and exploring its applications to various classes of topological groups.

Contribution

It establishes a new $z$-Minimality Criterion for dense subgroups and characterizes when the inner automorphism group of a connected Lie group is minimal.

Findings

Inner automorphism groups of connected Lie groups are minimal if the group is centre-free.

Connected locally compact nilpotent groups are $z$-minimal iff they are compact abelian.

Existence of a connected metabelian $z$-minimal Lie group that is neither compact nor abelian.

Abstract

By [6], a minimal group $G$ is called $z$-minimal if $G/Z(G)$ is minimal. In this paper, we present the $z$-Minimality Criterion for dense subgroups with some applications to topological matrix groups. For a locally compact group $G$, let $\operatorname{Inn}(G)$ be the group of all inner automorphisms of $G,$ endowed with the Birkhoff topology. Using a theorem by Goto [14], we obtain our main result which asserts that if $G$ is a connected Lie group and $H\in\{G/Z(G), \operatorname{Inn}(G)\},$ then $H$ is minimal if and only if it is centre-free and topologically isomorphic to $\operatorname{Inn}(G/Z(G)).$ In particular, if $G$ is a connected Lie group with discrete centre, then $\operatorname{Inn}(G)$ is minimal. We prove that a connected locally compact nilpotent group is $z$-minimal if and only if it is compact abelian. In contrast, we show that there exists a connected metabelian $z$-minimal Lie group that is neither compact nor abelian.