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 \(\mu\in\mathfrak{g}^*\), 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.