Abstract

ESPINOZA, Jesús  and  URIBE, Bernardo. Topological properties of spaces of projective unitary representations. Rev. acad. colomb. cienc. exact. fis. nat. [online]. 2016, vol.40, n.155, pp.337-352. ISSN 0370-3908.  https://doi.org/10.18257/raccefyn.317.

Let G be a compact and connected Lie group and PU(H) be the group of projective unitary operators on an infinite dimensional separable Hilbert space H endowed with the strong operator topology. We study the space homst(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 S¹-central extensions of G, and that each connected component has the group hom(G, S¹) for fundamental group and trivial higher homotopy groups. We study the conjugation map PU(H) → hom_st(G, PU(H)), F → FaF^-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 of finite paracompact dimension, local lifts to PU(H) do exist.

Keywords : Unitary Representation; Projective Unitary Representation.

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