Centralizers of Commuting Elements in Compact Lie Groups

TL;DR

This paper characterizes the moduli space of flat G-bundles over a torus for compact Lie groups, linking it to conjugacy classes of commuting pairs and analyzing centralizers using Dynkin diagram automorphisms.

Contribution

It provides a detailed analysis of centralizers of commuting elements in compact Lie groups and relates the moduli space to conjugacy classes, extending understanding of their structure.

Findings

Moduli space is homeomorphic to a product of tori mod Weyl group action.

Properties of centralizers are derived using diagram automorphisms.

There is a uniform bound on chains of commuting elements in G.

Abstract

The moduli space for a flat G-bundle over the two-torus is completely determined by its holonomy representation. When G is compact, connected, and simply connected, we show that the moduli space is homeomorphic to a product of two tori mod the action of the Weyl group, or equivalently to the conjugacy classes of commuting pairs of elements in G. Since the component group for a non-simply connected group is given by some finite dimensional subgroup in the centralizer of an n-tuple, we use diagram automorphisms of the extended Dynkin diagram to prove properties of centralizers of pairs of elements in G, followed by some explicit examples. We conclude by showing that as a result of a compact, connected, simply connected Lie group G having a finite number of subgroups, each conjugate to the centralizer of any element in G, that there is a uniform bound on an irredundant chain of commuting elements.