Beijing Institute of Mathematical Sciences and Applications

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

Robert McRae

Wednesday, November 27, 2024 4:30 PM - 5:30 PM

A3-2a-302

Zoom 815 762 8413 (BIMSA)

I will discuss recent joint works with Thomas Creutzig, Kenichi Shimizu, Harshit Yadav, and Jinwei Yang. Let $A$ be a commutative algebra in a braided monoidal category $C$. For example, $A$ could be a vertex operator algebra (VOA) extension of a VOA $V$ in a category $C$ of $V$-modules. First, assuming that $C$ is a finite braided tensor category, I will discuss conditions under which the category $C_A$ of $A$-modules in $C$ and its subcategory $C_A^{loc}$ of local modules inherit rigidity from $C$. These conditions are based on criteria of Etingof and Ostrik for $A$ to be an exact algebra in $C$. As an application, we show that if a simple non-negative integer-graded vertex operator algebra $A$ contains a strongly rational vertex operator subalgebra $V$, then $A$ is also strongly rational, without requiring the dimension of $A$ in the modular tensor category of $V$-modules to be non-zero. Second, assuming that $C$ is a Grothendieck-Verdier category (which means that $C$ admits a weaker duality structure than rigidity), I will discuss conditions under which $C$ inherits rigidity from $C_A^{loc}$. These conditions are motivated by free field-like VOA extensions $A$ of a vertex operator subalgebra $V$ where $A$ is often an indecomposable $V$-module. As an application, we show that the category of weight modules for the simple affine VOA of $sl_2$ at any admissible level is rigid.