Quantisation and nilpotent limits of Mishchenko-Fomenko subalgebras

Alexander Molev and Oksana Yakimova

Abstract

For any simple Lie algebra $\mathfrak{g}$ and an element $\mu \in {\mathfrak{g}}^{\ast }$, the corresponding commutative subalgebra ${\mathcal{A}}_{\mu }$ of $\mathcal{U}(\mathfrak{g})$ is defined as a homomorphic image of the Feigin-Frenkel centre associated with $\mathfrak{g}$. It is known that when $\mu$ is regular this subalgebra solves Vinberg's quantisation problem, as the graded image of ${\mathcal{A}}_{\mu }$ coincides with the Mishchenko-Fomenko subalgebra ${\bar{\mathcal{A}}}_{\mu }$ of $\mathcal{S}(\mathfrak{g})$. By a conjecture of Feigin, Frenkel and Toledano Laredo, this property extends to an arbitrary element $\mu$. We give sufficient conditions which imply the property for certain choices of $\mu$. In particular, this proves the conjecture in type C and gives a new proof in type A. We show that the algebra ${\mathcal{A}}_{\mu }$ is free in both cases and produce its generators in an explicit form. Moreover, we prove that in all classical types generators of ${\mathcal{A}}_{\mu }$ can be obtained via the canonical symmetrisation map from certain generators of ${\bar{\mathcal{A}}}_{\mu }$. The symmetrisation map is also used to produce free generators of nilpotent limits of the algebras ${\mathcal{A}}_{\mu }$ and give a positive solution of Vinberg's problem for these limit subalgebras.