Finite central extensions of type I

TL;DR

This paper proves that finite central extensions of certain type I Lie groups, specifically connected solvable ones, preserve the type I property, contrasting with the general case where this property may not be preserved.

Contribution

It establishes that finite central extensions of type I connected solvable Lie groups are also of type I, and explores algebraic hulls and subgroup intersections in these groups.

Findings

Finite central extensions of type I connected solvable Lie groups are also of type I.

Ad-algebraic hulls operate on these groups even when not simply connected.

Characterization of intersections of Euclidean subgroups containing a given central subgroup.

Abstract

Let $\mathbb{G}$ be a Lie group with solvable connected component and finitely-generated component group and $\alpha\in H^2(\mathbb{G},\mathbb{S}^1)$ a cohomology class. We prove that if $(\mathbb{G},\alpha)$ is of type I then the same holds for the finite central extensions of $\mathbb{G}$. In particular, finite central extensions of type-I connected solvable Lie groups are again of type I. This is by contrast with the general case, whereby the type-I property does not survive under finite central extensions. We also show that ad-algebraic hulls of connected solvable Lie groups operate on these even when the latter are not simply connected, and give a group-theoretic characterization of the intersection of all Euclidean subgroups of a connected, simply-connected solvable group $\mathbb{G}$ containing a given central subgroup of $\mathbb{G}$.