ИСТИНА

Let $\mathfrak{gl}_n$ be the Lie algebra of $n\times n$ matrices over a characteristic zero field $\Bbbk$ (one can take $\Bbbk=\mathbb R$ or $\mathbb C$). We denote by $S(\mathfrak{gl}_n)$ the Poisson algebra of polynomial functions on $\mathfrak{gl}_n^*$, and by $U\mathfrak{gl}_n$ the universal enveloping algebra of $\mathfrak{gl}_n$. By Poncar\'e-Birkhoff-Witt theorem, $U\mathfrak{gl}_n$ and $S(\mathfrak{gl}_n)$ are isomorphic as linear spaces; moreover, $S(\mathfrak{gl}_n)$ is isomorphic to the graded algebra $gr(U\mathfrak{gl}_n)$, associated with the standard filtration on $U\mathfrak{gl}_n$. Let $A$ be a Poisson-commutative subalgebra in $S(\mathfrak{gl}_n)$; one says that a commutative subalgebra $\hat A$ of $U\mathfrak{gl}_n$ is \textit{quantisation} of $A$, if its image under the natural projection $U\mathfrak{gl}_n\to gr(U\mathfrak{gl}_n)\cong S(\mathfrak{gl}_n)$ is equal to $A$. In my talk I will speak about the so-called ``argument shift'' subalgebras $A=A_\xi$ in $S(\mathfrak{gl}_n)$, see [1]. These subalgebras are generated by the iterated derivations of the central elements in this Poisson algebra in direction of a constant vector field $\xi$ on $\mathfrak{gl}_n^*$. Quantisations $\hat A_\xi$ of these algebras have been studied for the last 20 years; there exist several ways to define $\hat A_\xi$, most of them are related with the considerations of some infinite-dimensional Lie algebras, see [2], [3]. In my talk I will explain, how one can construct such quantisation of $A_\xi$ using as its generators iterated \textit{quasi-derivations} $\hat\xi$ of $U\mathfrak{gl}_n$. These operations were first defined in [4]; they are ``quantisations'' of the derivations on $S(\mathfrak{gl}_n)$ and verify an analog of the Leibniz rule with respect to the product in $U\mathfrak{gl}_n$. In fact, I will show that iterated quasiderivation of certain generating elements in $U\mathfrak{gl}_n$ are equal to the linear combinations of the elements, earlier constructed by Tarasov, see [5].