Braided commutative algebras over quantized enveloping algebras

Laugwitz, Robert; Walton, Chelsea

Abstract

We produce braided commutative algebras in braided monoidal categories by generalizing Davydov's full center construction of commutative algebras in centers of monoidal categories. Namely, we build braided commutative algebras in relative monoidal centers Z B pCq from algebras in B-central monoidal categories C, where B is an arbitrary braided monoidal category; Davydov's (and previous works of others) take place in the special case when B is the category of vector spaces Vect k over a field k. Since key examples of relative monoidal centers are suitable representation categories of quantized enveloping algebras, we supply braided commutative module algebras over such quantum groups. One application of our work is that we produce Morita invariants for algebras in B-central monoidal categories. Moreover, for a large class of B-central monoidal categories, our braided commutative algebras arise as a braided version of centralizer algebras. This generalizes the fact that centers of algebras in Vect k serve as Morita invariants. Many examples are provided throughout.

Citation

Laugwitz, R., & Walton, C. (2021). Braided commutative algebras over quantized enveloping algebras. Transformation Groups, 26, 957-993. https://doi.org/10.1007/s00031-020-09599-9

• Journal Article Type: Article
• Acceptance Date: May 17, 2020
• Online Publication Date: Jul 20, 2020
• Publication Date: 2021-09
• Deposit Date: Jun 12, 2020
• Publicly Available Date: Jul 21, 2021
• Journal: Transformation Groups
• Print ISSN: 1083-4362
• Electronic ISSN: 1531-586X
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 26
• Pages: 957-993
• DOI: https://doi.org/10.1007/s00031-020-09599-9
• Keywords: Geometry and Topology; Algebra and Number Theory
• Public URL: https://nottingham-repository.worktribe.com/output/4632251
• Publisher URL: https://link.springer.com/article/10.1007/s00031-020-09599-9
• Additional Information: This is a post-peer-review, pre-copyedit version of an article published in Transformation Groups. The final authenticated version is available online at: https://dx.doi.org/10.1007/s00031-020-09599-9 /> ; Correction to original version of paper: https://link.springer.com/article/10.1007%2Fs00031-021-09665-w