Quantization of the shift of argument subalgebras in type A

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.

18 pages

References in corpus (2)

• Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence
• On higher order Sugawara operators

Cited by in corpus (7)

• Quantisation and nilpotent limits of Mishchenko-Fomenko subalgebras
• Center of the quantum affine vertex algebra in type A
• Segal-Sugawara vectors for the Lie algebra of type $G_2$
• Spectra of Bethe subalgebras of $Y({\mathfrak{gl}}_n)$ in tame representations
• $L_{\infty }$-derivations and the argument shift method for deformation quantization algebras
• On the quantum argument shift method
• Dualities of Gaudin models with irregular singularities for general linear Lie (super)algebras