Trace class groups: the case of semi-direct products

TL;DR

This paper investigates trace class groups focusing on semi-direct products, establishing that such a product involving a semisimple Lie group and its Lie algebra is trace class only if the group is compact.

Contribution

It provides a characterization of when semi-direct products of semisimple Lie groups and their Lie algebras are trace class groups, extending previous classifications.

Findings

Semi-direct product of a semisimple Lie group and its Lie algebra is trace class iff the group is compact.

The paper advances understanding of trace class properties in the context of semi-direct product groups.

Theorem linking compactness of G to trace class property in semi-direct products.

Abstract

In this paper we continue the study of groups of trace class and consider in particular the case of semi-direct products. One of the highlights is the theorem saying that the semi-direct product of a semisimple Lie group G and its Lie algebra is a trace class group if and only if G is compact.