Centers of Braided Tensor Categories
Zhimin Liu, Shenglin Zhu

TL;DR
This paper explores the structure of centers in braided tensor categories, linking them to comodules over a cocommutative Hopf algebra, and applies findings to classify irreducible Yetter-Drinfeld modules over certain weak Hopf algebras.
Contribution
It establishes an isomorphism between the center of a braided category and comodules over Majid's automorphism braided group, generalizing previous results.
Findings
Center of $$ is isomorphic to left $B$-comodules in $$
Decomposition of $B$ induces a decomposition of comodules and subcategories
Explicit characterization of irreducible Yetter-Drinfeld modules over semisimple quasi-triangular weak Hopf algebras
Abstract
Let be a finite braided multitensor category. Let be Majid's automorphism braided group of , then is a cocommutative Hopf algebra in . We show that the center of is isomorphic to the category of left -comodules in , and the decomposition of into a direct sum of indecomposable -subcoalgebras leads to a decomposition of - into a direct sum of indecomposable -module subcategories. As an application, we present an explicit characterization of the structure of irreducible Yetter-Drinfeld modules over semisimple quasi-triangular weak Hopf algebras. Our results generalize those results on finite groups and on quasi-triangular Hopf algebras.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology
