2010/06/19 by Nairn, Kristen A.
#17B20 #22C05 (Primary) #22E46 (Secondary) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1006.3877
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.