QUANTISATION AND NILPOTENT LIMITS OF MISHCHENKO-FOMENKO SUBALGEBRAS

For any simple Lie algebra g and an element mu is an element of g*, the corresponding commutative subalgebra A(mu) of U(g) is defined as a homomorphic image of the Feigin-Frenkel centre associated with g. It is known that when mu is regular this subalgebra solves Vinberg's quantisation problem, as the graded image of A(mu) coincides with the Mishchenko-Fomenko subalgebra A (mu) over bar of S(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(mu) We show that the algebra 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 A(mu) can be obtained via the canonical symmetrisation map from certain generators of (A) over bar mu. The symmetrisation map is also used to produce free generators of nilpotent limits of the algebras A(mu) and to give a positive solution of Vinberg's problem for these limit subalgebras.