Piecewise Principal Coactions of Co-Commutative Hopf Algebras

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2014
• 1. Verfasser: Zieliński, B.
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут математики НАН України 2014
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/146612
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• 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.

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.