# Center at the critical level for centralizers in type $A$

## TL;DR

This paper explicitly constructs generators of the center of an affine vertex algebra at the critical level associated with the centralizer of a nilpotent element in type A, linking it to quantum shift algebras and universal enveloping algebra centers.

## Contribution

It provides an explicit construction of free generators of the center for the affine vertex algebra at the critical level related to centralizers in type A.

## Key findings

- Constructed explicit free generators of the center.
- Connected the generators to quantum shift of argument subalgebras.
- Reproduced known generators of the universal enveloping algebra center.

## Abstract

We consider the affine vertex algebra at the critical level associated with the centralizer of a nilpotent element in the Lie algebra $\mathfrak{gl}_N$. Due to a recent result of Arakawa and Premet, the center of this vertex algebra is an algebra of polynomials. We construct a family of free generators of the center in an explicit form. As a corollary, we obtain generators of the corresponding quantum shift of argument subalgebras and recover free generators of the center of the universal enveloping algebra of the centralizer produced earlier by Brown and Brundan.