Piecewise Principal Coactions of Co-Commutative Hopf Algebras
Principal comodule algebras can be thought of as objects representing principal bundles in non-commutative geometry. A crucial component of a principal comodule algebra is a strong connection map. For some applications it suffices to prove that such a map exists, but for others, such as computing th...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2014 |
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| Format: | Artikel |
| Sprache: | English |
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Інститут математики НАН України
2014
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/146612 |
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| Zitieren: | Piecewise Principal Coactions of Co-Commutative Hopf Algebras / B. Zieliński // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 23 назв. — англ. |
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Zieliński, B. 2019-02-10T10:02:30Z 2019-02-10T10:02:30Z 2014 Piecewise Principal Coactions of Co-Commutative Hopf Algebras / B. Zieliński // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58B32; 16T05 DOI:10.3842/SIGMA.2014.088 https://nasplib.isofts.kiev.ua/handle/123456789/146612 Principal comodule algebras can be thought of as objects representing principal bundles in non-commutative geometry. A crucial component of a principal comodule algebra is a strong connection map. For some applications it suffices to prove that such a map exists, but for others, such as computing the associated bundle projectors or Chern-Galois characters, an explicit formula for a strong connection is necessary. It has been known for some time how to construct a strong connection map on a multi-pullback comodule algebra from strong connections on multi-pullback components, but the known explicit general formula is unwieldy. In this paper we derive a much easier to use strong connection formula, which is not, however, completely general, but is applicable only in the case when a Hopf algebra is co-commutative. Because certain linear splittings of projections in multi-pullback comodule algebras play a crucial role in our construction, we also devote a significant part of the paper to the problem of existence and explicit formulas for such splittings. Finally, we show example application of our work. This paper is a contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rief fel. The full collection is available at http://www.emis.de/journals/SIGMA/Rieffel.html. The author is grateful to Piotr M. Hajac for helpful discussions. The author would also like to thank the referees for helpful suggestions. This work was partially supported by the NCN-grant 2012/06/M/ST1/00169. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Piecewise Principal Coactions of Co-Commutative Hopf Algebras Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Piecewise Principal Coactions of Co-Commutative Hopf Algebras |
| spellingShingle |
Piecewise Principal Coactions of Co-Commutative Hopf Algebras Zieliński, B. |
| title_short |
Piecewise Principal Coactions of Co-Commutative Hopf Algebras |
| title_full |
Piecewise Principal Coactions of Co-Commutative Hopf Algebras |
| title_fullStr |
Piecewise Principal Coactions of Co-Commutative Hopf Algebras |
| title_full_unstemmed |
Piecewise Principal Coactions of Co-Commutative Hopf Algebras |
| title_sort |
piecewise principal coactions of co-commutative hopf algebras |
| author |
Zieliński, B. |
| author_facet |
Zieliński, B. |
| publishDate |
2014 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
Principal comodule algebras can be thought of as objects representing principal bundles in non-commutative geometry. A crucial component of a principal comodule algebra is a strong connection map. For some applications it suffices to prove that such a map exists, but for others, such as computing the associated bundle projectors or Chern-Galois characters, an explicit formula for a strong connection is necessary. It has been known for some time how to construct a strong connection map on a multi-pullback comodule algebra from strong connections on multi-pullback components, but the known explicit general formula is unwieldy. In this paper we derive a much easier to use strong connection formula, which is not, however, completely general, but is applicable only in the case when a Hopf algebra is co-commutative. Because certain linear splittings of projections in multi-pullback comodule algebras play a crucial role in our construction, we also devote a significant part of the paper to the problem of existence and explicit formulas for such splittings. Finally, we show example application of our work.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146612 |
| citation_txt |
Piecewise Principal Coactions of Co-Commutative Hopf Algebras / B. Zieliński // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 23 назв. — англ. |
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AT zielinskib piecewiseprincipalcoactionsofcocommutativehopfalgebras |
| first_indexed |
2025-12-02T08:43:44Z |
| last_indexed |
2025-12-02T08:43:44Z |
| _version_ |
1850861941704097792 |