Seminar talk, 15 March 2023

Speaker: Georgy Sharygin

Title: Quasiderivations and commutative subalgebras of the algebra $U{\mathfrak {gl}}_{n}$

Abstract:
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 $\mathbb {C}$); let $S({\mathfrak {gl}}_{n})$ be the Poisson algebra of polynomial functions on ${\mathfrak {gl}}_{n}^{*}$, and $U{\mathfrak {gl}}_{n}$ the universal enveloping algebra of ${\mathfrak {gl}}_{n}$. By Poncar\'e-Birkhoff-Witt theorem $S({\mathfrak {gl}}_{n})$ is isomorphic to the graded algebra $gr(U{\mathfrak {gl}}_{n})$, associated with the order filtration on $U{\mathfrak {gl}}_{n}$. Let $A\subseteq S({\mathfrak {gl}}_{n})$ be a Poisson-commutative subalgebra; one says that a commutative subalgebra ${\hat {A}}\subseteq U{\mathfrak {gl}}_{n}$ is a \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 $S({\mathfrak {gl}}_{n})$, generated by the iterated derivations of central elements in $S({\mathfrak {gl}}_{n})$ by a constant vector field $\xi$. There exist several ways to define a quantisation of $A_{\xi }$, 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 $A_{\xi }$ using as its generators iterated \textit{quasi-derivations} ${\hat {\xi }}$ of $U{\mathfrak {gl}}_{n}$. These operations are "quantisations" of the derivations on $S({\mathfrak {gl}}_{n})$ and verify an analog of the Leibniz rule. In fact, I will show that iterated quasiderivation of certain generating elements in $U{\mathfrak {gl}}_{n}$ are equal to the linear combinations of the elements, earlier constructed by Tarasov.