Quantization of the shift of argument subalgebras in type A

TL;DR

This paper proves that for type A Lie algebras, the quantization of shift of argument subalgebras can be extended from regular to all elements, confirming a conjecture by Feigin, Frenkel, and Toledano Laredo.

Contribution

It establishes that the quantization property of shift of argument subalgebras in type A Lie algebras holds for all elements, not just regular ones, using explicit constructions from affine vertex algebra centers.

Findings

Quantization extends to all elements in type A Lie algebras.

Constructs explicit generators of the affine vertex algebra center.

Confirms a conjecture by Feigin, Frenkel, and Toledano Laredo.

Abstract

Given a simple Lie algebra $\mathfrak{g}$ and an element $\mu\in\mathfrak{g}^*$, 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.