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 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 $\overline{ \mathcal{ A}}_{ \mu}$. 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.
Categories: math.RT