Representations of the quantum affine vertex algebra associated with the trigonometric $R$-matrix of type $A$

Abstract: One important problem in the vertex algebra theory is to associate certain vertex algebra-like objects, the quantum vertex algebras, to various classes of quantum groups, such as quantum affine algebras or double Yangians. In this talk, I will discuss this problem in the context of Etingof--Kazhdan's quantum affine vertex algebra $\mathcal{V}^c(\mathfrak{gl}_N)$ associated with the trigonometric $R$-matrix of type $A$. The main focus will be on the explicit description of the center of $\mathcal{V}^c(\mathfrak{gl}_N)$ at the critical level $c=-N$ and, furthermore, on the connection between certain classes of $\mathcal{V}^c(\mathfrak{gl}_N)$-modules and representation theories of the quantum affine algebra of type $A$ and the orthogonal twisted $h$-Yangian. The talk is in part based on the joint works with Alexander Molev and Lucia Bagnoli.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar