Topological properties of spaces of projective unitary representations

Abstract

Let G be a compact and connected Lie group and PU(H) be the group of projective unitary operators on a separable Hilbert space H endowed with the strong operator topology. We study the space Hom_{st}(G,PU(H)) of continuous homomorphisms from G to PU(H) which are stable, namely the homomorphisms whose induced representation contains each irreducible representation an infinitely number of times. We show that the connected components of Hom_{st}(G,PU(H)) are parametrized by the isomorphism classes of S^1- entral extensions of G, and that each connected component has the group Hom(G,S^1) for fundamental group and trivial higher homotopy groups. We study the conjugation map PU(H) -> Hom_{st}(G,PU(H)), F |-> F\alpha F^{-1}, we show that it has no local cross sections and we prove that for a map B -> Hom_{st}(G,PU(H)) with B paracompact, local lifts to PU(H) do exist. © 2016. Acad. Colomb. Cienc. Ex. Fis. Nat. All rights reserved.