PreprintQuantisation and nilpotent limits of MishchenkoFomenko subalgebrasAlexander Molev and Oksana YakimovaAbstractFor 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 FeiginFrenkel 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 MishchenkoFomenko 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. This paper is available as a pdf (424kB) file.
