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.