8th Croatian Mathematical Congress

Classification of irreducible modules for the affine vertex algebra L−2n+12(sl2n) in certain categories

2 Jul 2024, 17:50

20m

D8 (School of Applied Mathematics and Informatics, J. J. Strossmayer University of Osijek)

Talk ALG: Algebra Algebra

Speaker

Ivana Vukorepa

Description

We investigate the representation theory of simple affine vertex algebra $L_k(\mathfrak{g})$ at special non-admissible levels $k_n=-\frac{2n+1}{2}$ for $\mathfrak{g}={\mathfrak{sl}}_{2n}$. We classify irreducible $L_{k_n}({\mathfrak{sl}}_{2n})$-modules in category $K{L}_{k_n}({\mathfrak{sl}}_{2n})$ and prove that $K{L}_{k_n}({\mathfrak{sl}}_{2n})$ is a semi-simple, rigid braided tensor category.
In addition, we present a new method for proving simplicity of quotients of universal affine vertex algebras $V^{k_n}({\mathfrak{sl}}_{2n})$. We use this result to prove that in the case $n=3$ a maximal ideal is generated by one singular vector of conformal weight 4. As a byproduct, we classify irreducible modules in the category $\mathcal{O}$ for $L_{-7/2}({\mathfrak{sl}}_6)$.
The talk is based on joint papers with D. Adamović, T. Creutzig and O. Perše.