The relative monoidal center and tensor products of monoidal categories

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

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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
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
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Depositing User: LivePure Connector
Date Deposited: 04 Oct 2019 11:30
Last Modified: 04 Sep 2020 23:52
DOI: 10.1142/S0219199719500688

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