Quantization of the shift of argument subalgebras in type A

Vyacheslav Futorny and Alexander Molev

Abstract

Given a simple Lie algebra $\mathfrak{g}$ and an element $\mu \in {\mathfrak{g}}^{\ast }$, the corresponding shift of argument subalgebra of $\text{S}(\mathfrak{g})$ is Poisson commutative. In the case where $\mu$ is regular, this subalgebra is known to admit a quantization, that is, it can be lifted to a commutative subalgebra of $\text{U}(\mathfrak{g})$. We show that if $\mathfrak{g}$ is of type $A$, then this property extends to arbitrary $\mu$, thus proving a conjecture of Feigin, Frenkel and Toledano Laredo. The proof relies on an explicit construction of generators of the center of the affine vertex algebra at the critical level.