Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology
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...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2023 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2023
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212025 |
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| 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 |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862603589498699776 |
|---|---|
| author | Chirvasitu, Alexandru Peng, Jun |
| author_facet | Chirvasitu, Alexandru Peng, Jun |
| citation_txt | Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology. Alexandru Chirvasitu and Jun Peng. SIGMA 19 (2023), 106, 28 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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).
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| first_indexed | 2026-03-14T03:46:06Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212025 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T03:46:06Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Chirvasitu, Alexandru Peng, Jun 2026-01-23T10:07:37Z 2023 Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology. Alexandru Chirvasitu and Jun Peng. SIGMA 19 (2023), 106, 28 pages 1815-0659 2020 Mathematics Subject Classification: 22E65; 17B65; 58B25; 22E41; 57N35; 46L05; 16H05; 16D60; 16K20 arXiv:2211.11429 https://nasplib.isofts.kiev.ua/handle/123456789/212025 https://doi.org/10.3842/SIGMA.2023.106 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). This work is partially supported by NSF grant DMS-2001128. We are grateful for valuable pointers to the literature from Harl-Hermann Neeb, as well as the anonymous referees for insightful comments and suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology Article published earlier |
| spellingShingle | Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology Chirvasitu, Alexandru Peng, Jun |
| title | Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology |
| title_full | Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology |
| title_fullStr | Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology |
| title_full_unstemmed | Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology |
| title_short | Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology |
| title_sort | manifolds of lie-group-valued cocycles and discrete cohomology |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212025 |
| work_keys_str_mv | AT chirvasitualexandru manifoldsofliegroupvaluedcocyclesanddiscretecohomology AT pengjun manifoldsofliegroupvaluedcocyclesanddiscretecohomology |