ИСТИНА

Argument shift method is a well-known method of obtaining Poisson-commutative subalgebras in the algebra of smooth functions C∞(X) of a Poisson manifold X. It is based on the observation that if a vector field ξ and the Poisson bivector π on X verify the equations Lξπ ̸= 0, L2ξπ = 0, (here Lξ denotes the Lie derivative), then the elements Lpξ(f) Poisson-commute with each other for all p ≥ 0 and all smooth Casimir functions f . For instance when X = g∗ (the dual space of a Lie algebra) and the Poisson structure is the canonical Lie-Poisson structure on g∗, we can take the vector field ξ to be constant with respect to affine coordinates on g∗. This situation was first considered by Mischenko and Fomenko in [1]; for a generic vector field ξ this method gives a maximal Poisson-commutative subalgebra Aξ in the symmetric algebra S(g); Aξ is often called Mischenko-Fomenko algebra. In 1991 E.B. Vinberg asked, whether it was possible to find a commutative subalgebra ˆξ in the universal enveloping algebra Ug, such that the image of ˆξ in Sg under the canonical isomorphism of associated graded algebra of Ug and Sg would be equal toAξ. This question has been solved by various people, the best known construction of the quantum Mischenko-Fomenko algebra ˆξ being that of Rybnikov, see [2]. However, finding an element in Rybnikov’s algebra that will correspond to a particular ξp(f) in Aξ is not easy. In my talk I will describe a method of quantising the elements ξp(f) (i.e. raising it to ˆξ ⊂ Ug for g = gld. It is based on a systematic use of the quasiderivation operation of Gurevich, Pyatov and Saponov (see [3]) ˆ on 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.