Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2023
• ISSN: 1815-0659
• Main Authors: Chirvasitu, Alexandru, Peng, Jun
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2023
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/212025
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology. Alexandru Chirvasitu and Jun Peng. SIGMA 19 (2023), 106, 28 pages

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