ИСТИНА

Let X be a Poisson manifold. Argument shift method is a well-known method of obtaining commutative subalgebras in the Poisson algebra C∞(X). It is based on the observation that if a vector field ξ and the Poisson bivector pi verify the following equalities Lξpi 6= 0, L2ξpi = 0, where Lξ denotes the Lie derivative in direction of ξ, then the elements Lpξ(f) Poisson- commute with each other for all p ≥ 0 and all Casimir functions f . An important particular case of this construction is when X = g∗ is equal to the dual space of a Lie algebra and the Poisson structure is equal to the canonical Lie-Poisson structure on g∗ (i.e. is induced from the Lie algebra structure on g) and the vector field is constant with respect to affine coordinates on g∗. This situation was first considered by Mischenko and Fomenko in 1 In this case for a generic vector field ξ the method gives a maximal commutative subalgebra in the symmetric algebra S(g), often called Mischenko-Fomenko algebra. In 1991 E. B. Vinberg asked the following question: Is it possible to find a com- mutative subalgebra Aξ in the universal enveloping algebra Ug of a Lie algebra, such that its image in Sg under the canonical isomorphism of associated graded algebra of Ug and Sg would be equal to Mischenko-Fomenko algebra? This question was solved by various people, the best known construction of the quantum Mischenko-Fomenko algebra Aξ being that of Rybnikov, see 2 However, finding an element in Rybnikov’s algebra that will correspond to a particular ξp(f) in Mischenko-Fomenko algebra is not a simple task. In my talk I will describe a method of quantising the element ξp(f) (i.e. raising it to Aξ ⊂ Ug for g = gld. It is based on a systematic use of the quasiderivation operation of Gurevich and Saponov, (see 3.) ˆon Ugld in the stead of usual directional derivative ξ (we assume that the coefficients of ˆ coincide with those of ξ). Namely, we can prove the following result Theorem. Let ˆ∈ Ugld be a central element, p ≥ 0; then the element ˆ ( ˆ is in the quantum Mischenko-Fomenko algebra Aξ. In particular, all such elements commute with each other. The talk is based on a joint work with Yasushi Ikeda, arXiv:2307.15952.