Seminar talk, 15 March 2023

Title: Quasiderivations and commutative subalgebras of the algebra ${\displaystyle U{\mathfrak {gl}}_{n}}$
Let ${\displaystyle {\mathfrak {gl}}_{n}}$ be the Lie algebra of ${\displaystyle n\times n}$ matrices over a characteristic zero field ${\displaystyle \Bbbk }$ (one can take ${\displaystyle \Bbbk =\mathbb {R} }$ or ${\displaystyle \mathbb {C} }$); let ${\displaystyle S({\mathfrak {gl}}_{n})}$ be the Poisson algebra of polynomial functions on ${\displaystyle {\mathfrak {gl}}_{n}^{*}}$, and ${\displaystyle U{\mathfrak {gl}}_{n}}$ the universal enveloping algebra of ${\displaystyle {\mathfrak {gl}}_{n}}$. By Poincaré-Birkhoff-Witt theorem ${\displaystyle S({\mathfrak {gl}}_{n})}$ is isomorphic to the graded algebra ${\displaystyle gr(U{\mathfrak {gl}}_{n})}$, associated with the order filtration on ${\displaystyle U{\mathfrak {gl}}_{n}}$. Let ${\displaystyle A\subseteq S({\mathfrak {gl}}_{n})}$ be a Poisson-commutative subalgebra; one says that a commutative subalgebra ${\displaystyle {\hat {A}}\subseteq U{\mathfrak {gl}}_{n}}$ is a quantisation of ${\displaystyle A}$, if its image under the natural projection ${\displaystyle U{\mathfrak {gl}}_{n}\to gr(U{\mathfrak {gl}}_{n})\cong S({\mathfrak {gl}}_{n})}$ is equal to ${\displaystyle A}$. In my talk I will speak about the so-called "argument shift" subalgebras ${\displaystyle A=A_{\xi }}$ in ${\displaystyle S({\mathfrak {gl}}_{n})}$, generated by the iterated derivations of central elements in ${\displaystyle S({\mathfrak {gl}}_{n})}$ by a constant vector field ${\displaystyle \xi }$. There exist several ways to define a quantisation of ${\displaystyle A_{\xi }}$, most of them are related with the considerations of some infinite-dimensional Lie algebras. In my talk I will explain, how one can construct such quantisation of ${\displaystyle A_{\xi }}$ using as its generators iterated quasi-derivations ${\displaystyle {\hat {\xi }}}$ of ${\displaystyle U{\mathfrak {gl}}_{n}}$. These operations are "quantisations" of the derivations on ${\displaystyle S({\mathfrak {gl}}_{n})}$ and verify an analog of the Leibniz rule. In fact, I will show that iterated quasiderivation of certain generating elements in ${\displaystyle U{\mathfrak {gl}}_{n}}$ are equal to the linear combinations of the elements, earlier constructed by Tarasov.