### Abstract

Given a Hopf algebra H, we study modules and bimodules over an algebra A that carry an H-action, as well as their morphisms and connections. Bimodules naturally arise when considering noncommutative analogues of tensor bundles. For quasitriangular Hopf algebras and bimodules with an extra quasi-commutativity property we induce connections on the tensor product over A of two bimodules from connections on the individual bimodules. This construction applies to arbitrary connections, i.e. not necessarily Hequivariant ones, and further extends to the tensor algebra generated by a bimodule and its dual. Examples of these noncommutative structures arise in deformation quantization via Drinfeld twists of the commutative differential geometry of a smooth manifold, where the Hopf algebra H is the universal enveloping algebra of vector fields (or a finitely generated Hopf subalgebra). We extend the Drinfeld twist deformation theory of modules and algebras to morphisms and connections that are not necessarily H-equivariant. The theory canonically lifts to the tensor product structure.

Original language | English |
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Pages (from-to) | 513-612 |

Number of pages | 100 |

Journal | Advances in Theoretical and Mathematical Physics |

Volume | 18 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2014 |

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Mathematics(all)

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## Cite this

*Advances in Theoretical and Mathematical Physics*,

*18*(3), 513-612. https://doi.org/10.4310/ATMP.2014.v18.n3.a1