Beijing Institute of Mathematical Sciences and Applications

Lin Zhe Huang , Zheng Wei Liu , Sébastien Palcoux , Yi Long Wang , Jin Song Wu

Kenichi Shimizu

Thursday, December 19, 2024 2:30 PM - 3:30 PM

A3-4-101

Zoom 637 734 0280 (BIMSA)

Given a commutative algebra $A$ in a braided monoidal category $C$, the category $C_A^{loc}$ of local $A$-modules in $C$ is defined as a certain full subcategory of the category of $A$-bimodules in $C$. As has been pointed out by Pareigis, provided that $C$ admits coequalizers and the tensor product of $C$ preserves them, the category $C_A^{loc}$ has a natural structure of a braided monoidal category inherited from that of $C$. As it later turned out, the category of local modules is related to representations of an extension of a vertex operator algebra. We are therefore interested in knowing when the category of local modules has nice properties. In this talk, I will introduce a criterion for $C_A^{loc}$ to be rigid monoidal. As an application, $C_A^{loc}$ is a braided finite tensor category if $C$ is a braided finite tensor category and the category of $A$-bimodules is a finite tensor category (or, equivalently, $A$ is an indecomposable exact commutative algebra in $C$). If, in addition, $C$ is non-degenerate and $A$ is symmetric Frobenius, then $C_A^{loc}$ is a modular tensor category in the sense of Lyubashenko. I will also discuss the Witt equivalence of non-degenerate braided finite tensor categories and relevant questions.