Kernels of Linear Representations of Lie Groups, Locally Compact Groups, and Pro-Lie Groups

TL;DR

This paper investigates the kernel of all ordinary representations of topological groups, especially connected pro-Lie groups, revealing their structure and limitations, with explicit classifications for certain classes of Lie groups.

Contribution

It characterizes the kernel intersection KO(G) for connected pro-Lie groups and explicitly determines KO(C) for classes of connected and almost connected Lie groups.

Findings

KO(G) is contained in the center for connected pro-Lie groups

KO(C) for certain classes consists of all compactly generated abelian Lie groups

The dimension of KO(G) is bounded by the continuum for locally compact connected groups

Abstract

For a topological group G the intersection KO(G) of all kernels of ordinary representations is studied. We show that KO(G) is contained in the center of G if G is a connected pro-Lie group. The class KO(C) is determined explicitly if C is the class ConnLie of connected Lie groups or the class almConnLie of almost connected Lie groups: in both cases, it consists of all compactly generated abelian Lie groups. Every compact abelian group and every connected abelian pro-Lie group occurs as KO(G) for some connected pro-Lie group G. However, the dimension of KO(G) is bounded by the cardinality of the continuum if G is locally compact and connected. Examples are given to show that KO(C) becomes complicated if C contains groups with infinitely many connected components.