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arXiv · 1403.7818

Piecewise Principal Coactions of Co-Commutative Hopf Algebras

Bartosz Zieliński
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Abstract

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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Bartosz Zieliński. 2014-03-30. Piecewise Principal Coactions of Co-Commutative Hopf Algebras. https://doi.org/10.3842/sigma.2014.088

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