Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology

• Опубліковано в: : Symmetry, Integrability and Geometry: Methods and Applications
• Дата: 2023
• Автори: Chirvasitu, Alexandru, Peng, Jun
• Формат: Стаття
• Мова: Англійська
• Опубліковано: Інститут математики НАН України 2023
• Онлайн доступ: https://nasplib.isofts.kiev.ua/handle/123456789/212025
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology. Alexandru Chirvasitu and Jun Peng. SIGMA 19 (2023), 106, 28 pages

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 𝛫 ≤ 𝑈 invariant. We define the spaces 𝛫𝑍ⁿ(𝐺, 𝑈) of 𝛫-relative continuous cocycles as those maps 𝐺ⁿ → 𝑈 whose coboundary is a 𝛫-valued (𝑛 + 1)-cocycle; this applies to possibly non-abelian 𝑈, in which case 𝑛 = 1. We show that the 𝛫𝑍ⁿ(𝐺, 𝑈) are analytic submanifolds of the spaces 𝐶(𝐺ⁿ, 𝑈) of continuous maps 𝐺ⁿ → 𝑈 and that they decompose as disjoint unions of fiber bundles over manifolds of 𝛫-valued cocycles. Applications include: (a) the fact that 𝑍ⁿ(𝐺, 𝑈) ⊂ 𝐶(𝐺ⁿ, 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 𝐻ⁿ(𝐺, 𝑈) are discrete; (c) for unital 𝐶*-algebras 𝐴 and 𝐵 with 𝐴 finite-dimensional the space of morphisms 𝐴 → 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).