## Quantisation and nilpotent limits of Mishchenko-Fomenko subalgebras

### Alexander Molev and Oksana Yakimova

#### Abstract

For any simple Lie algebra $$\mathfrak{g}$$ and an element $$\mu\in\mathfrak{g}^*$$, 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.

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 Tuesday, November 14, 2017