Centralizers of Commuting Elements in Compact Lie Groups
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.
Cite
@article{arxiv.1006.3877,
title = {Centralizers of Commuting Elements in Compact Lie Groups},
author = {Kristen A. Nairn},
journal= {arXiv preprint arXiv:1006.3877},
year = {2010}
}