The relative monoidal center and tensor products of monoidal categories

Laugwitz, Robert

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.

Citation

Laugwitz, R. (2020). The relative monoidal center and tensor products of monoidal categories. Communications in Contemporary Mathematics, 22(8), https://doi.org/10.1142/s0219199719500688

• Journal Article Type: Article
• Acceptance Date: Jul 24, 2019
• Online Publication Date: Aug 12, 2019
• Publication Date: 2020-12
• Deposit Date: Oct 8, 2019
• Publicly Available Date: Aug 13, 2020
• Journal: Communications in Contemporary Mathematics
• Print ISSN: 0219-1997
• Electronic ISSN: 1793-6683
• Publisher: World Scientific
• Peer Reviewed: Peer Reviewed
• Volume: 22
• Issue: 8
• Article Number: 1950068
• DOI: https://doi.org/10.1142/s0219199719500688
• Keywords: Monoidal center; Categorical modules; braided monoidal categories; Relative tensor product
• Public URL: https://nottingham-repository.worktribe.com/output/2782666
• Publisher URL: https://www.worldscientific.com/doi/abs/10.1142/S0219199719500688