Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 106, 28 pages      arXiv:2211.11429      https://doi.org/10.3842/SIGMA.2023.106

Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology

Alexandru Chirvasitu and Jun Peng
Department of Mathematics, University at Buffalo, Buffalo, NY 14260-2900, USA

Received June 18, 2023, in final form December 01, 2023; Published online December 24, 2023

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 ${}_KZ^n(G,U)$ 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 ${}_KZ^n(G,U)$ 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 $C^*$-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 $C^*$ algebras (the last recovering a result of Martin's).

Key words: Banach Lie group; Lie algebra; group cohomology; cocycle; coboundary; infinite-dimensional manifold; immersion; analytic; $C^*$-algebra; unitary group; Banach algebra; semisimple; Jacobson radical.

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