Quantisation and nilpotent limits of Mishchenko–Fomenko subalgebras

Abstract:

For any simple Lie algebra $\mathfrak {g}$ and an element $\mu \in \mathfrak {g}^*$, 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 $\overline {\mathcal {A}}_{\mu }$ 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 on $\mu$ which imply the property. 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 $\overline {\mathcal {A}}_{\mu }$. The symmetrisation map is also used to produce free generators of nilpotent limits of the algebras $\mathcal {A}_{\mu }$ and to give a positive solution of Vinberg’s problem for these limit subalgebras.