Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology

Abstract

Consider a compact group $G$ acting on a real or complex Banach Lie group $U$, by automorphisms in the relevant category, and leaving a central subgroup $K\le U$ invariant. We define the spaces ${}_K{Z}^n(G,U)$ of $K$-relative continuous cocycles as those maps $G^n\to U$ whose coboundary is a $K$-valued $(n+1)$-cocycle; this applies to possibly non-abelian $U$, in which case $n=1$. We show that the ${}_K{Z}^n(G,U)$ are analytic submanifolds of the spaces $C(G^n,U)$ of continuous maps $G^n\to U$ and that they decompose as disjoint unions of fiber bundles over manifolds of $K$-valued cocycles. Applications include: (a) the fact that $Z^n(G,U)\subset C(G^n,U)$ is an analytic submanifold and its orbits under the adjoint of the group of $U$-valued $(n-1)$-cochains are open; (b) hence the cohomology spaces $H^n(G,U)$ are discrete; (c) for unital $C^{\ast }$-algebras $A$ and $B$ with $A$ finite-dimensional the space of morphisms $A\to B$ is an analytic manifold and nearby morphisms are conjugate under the unitary group $U(B)$; (d) the same goes for $A$ and $B$ Banach, with $A$ finite-dimensional and semisimple; (e) and for spaces of projective representations of compact groups in arbitrary $C^{\ast }$ algebras (the last recovering a result of Martin's).

final version, to appear in SIGMA; 26 pages + references

References in corpus (2)

• Cohomology of the space of commuting n-tuples in a compact Lie group
• Fundamental Groups of Commuting Elements in Lie Groups