Topological properties of spaces of projective unitary representations

TL;DR

This paper investigates the topological structure of the space of stable projective unitary representations of a compact Lie group, revealing its connected components, fundamental groups, and properties of the conjugation map.

Contribution

It characterizes the connected components of the space of stable homomorphisms via $S^1$-central extensions and analyzes the topological properties of the conjugation map.

Findings

Connected components correspond to $S^1$-central extensions.

Each component has fundamental group $hom(G,S^1)$.

Conjugation map has no local cross sections but admits local lifts under certain conditions.

Abstract

Let $G$ be a compact and connected Lie group and $PU(\mathcal H)$ be the group of projective unitary operators on a separable Hilbert space $\mathcal H$ endowed with the strong operator topology. We study the space $hom_{st}(G, PU(\mathcal H))$ of continuous homomorphisms from $G$ to $PU(\mathcal H)$ 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 $hom_{st}(G, PU(\mathcal H))$ are parametrized by the isomorphism classes of $S^1$-central 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 $PU(\mathcal H) \to hom_{st}(G, PU(\mathcal H))$, $F \mapsto F\alpha F^{-1}$, we show that it has no local cross sections and we prove that for a map $B \to hom_{st}(G, PU(\mathcal H))$ with $B$ paracompact of finite covering dimension, local lifts to $PU(\mathcal H)$ do exist.