Centralizers of Commuting Elements in Compact Lie Groups
Kris A Nairn*
College of St. Benedict, 37 South College Avenue, St. Joseph, MN 56374, USA
- *Corresponding Author:
- Kris A Nairn
College of St. Benedict, 37 South College Avenue
St. Joseph, MN 56374, USA
E-mail: [email protected]
Received Date: September 30, 2016; Accepted Date: October 26, 2016; Published Date: November 04, 2016
Citation: Nairn KA (2016) Centralizers of Commuting Elements in Compact Lie Groups. J Generalized Lie Theory Appl 10: 246. doi:10.4172/1736-4337.1000246
Copyright: © 2016 Nairn KA. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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.