Geometric Representation Theory Seminar

Robert McRae

Assistant Professor

YMSC

Tsinghua University

I study the representation theory of algebraic structures that arise in two-dimensional conformal quantum field theories, especially vertex operator algebras, affine Lie (super)algebras, and the Virasoro Lie algebra. More specifically, I am interested in the existence, properties, and structure of braided tensor categories of modules for these algebras.

Organizers

Lin Chen, Will Donovan, Penghui Li, Peng Shan, Changjian Su, Wenbin Yan

Speaker

Robert McRae (YMSC)

Time

Friday, 15:30 - 16:30

Feb 21, 2025

Venue

B626, Shuangqing Complex Building

Tensor structure on the Kazhdan-Lusztig category of affine $\mathfrak{sl}_2$ at admissible levels

For a simple Lie algebra $\mathfrak{g}$ and a level $k$, the Kazhdan-Lusztig category $KL_k(\mathfrak{g})$ is the category of finite-length modules for the affine Lie algebra of $\mathfrak{g}$ at level $k$ whose composition factors have highest weights which are dominant integral for the subalgebra $\mathfrak{g}$. In this talk, I will discuss joint work with Jinwei Yang showing that $KL_k(\mathfrak{sl}_2)$ is a non-rigid braided tensor category when $k=-2+\frac{p}{q}$ is admissible, and that there is an exact and essentially surjective (but not quite full or faithful) tensor functor from $KL_k(\mathfrak{sl}_2)$ to the non-semisimple category of finite dimensional weight modules for Lusztig's big quantum group of $\mathfrak{sl}_2$ at the root of unity $e^{\pi i q/p}$. I will also discuss prospects for extending such results to higher rank $\mathfrak{g}$.