Laugwitz, Robert
(2020)
*The relative monoidal center and tensor products of monoidal categories.*
Communications in Contemporary Mathematics, 22 (8).
ISSN 0219-1997

## Abstract

This paper develops a theory of monoidal categories relative to a braided monoidal category, called augmented monoidal categories. For such categories, balanced bimodules are defined using the formalism of balanced functors. It is shown that there exists a monoidal structure on the relative tensor product of two augmented monoidal categories which is Morita dual to a relative version of the monoidal center. In examples, a category of locally finite weight modules over a quantized enveloping algebra is equivalent to the relative monoidal center of modules over its Borel part. A similar result holds for small quantum groups, without restricting to locally finite weight modules. More generally, for modules over bialgebras inside a braided monoidal category, the relative center is shown to be equivalent to the category of Yetter-Drinfeld modules inside the braided category. If the braided category is given by modules over a quasitriangular Hopf algebra, then the relative center corresponds to modules over a braided version of the Drinfeld double (i.e. the double bosonization in the sense of Majid) which are locally finite for the action of the dual.

Item Type: | Article |
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Uncontrolled Keywords: | braided monoidal categories,categorical modules,monoidal center,relative tensor product,mathematics(all),applied mathematics ,/dk/atira/pure/subjectarea/asjc/2600 |

Faculty \ School: | Faculty of Science > School of Mathematics |

Related URLs: | |

Depositing User: | LivePure Connector |

Date Deposited: | 04 Oct 2019 11:30 |

Last Modified: | 24 Sep 2022 04:54 |

URI: | https://ueaeprints.uea.ac.uk/id/eprint/72481 |

DOI: | 10.1142/S0219199719500688 |

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