Seminar talk, 15 March 2023

Speaker: Georgy Sharygin

Title: Quasiderivations and commutative subalgebras of the algebra $U{𝔤𝔩}_n$

Abstract:
Let ${𝔤𝔩}_n$ be the Lie algebra of $n\times n$ matrices over a characteristic zero field $𝕜$ (one can take $𝕜=ℝ$ or $ℂ$); let $S({𝔤𝔩}_n)$ be the Poisson algebra of polynomial functions on ${𝔤𝔩}_n^{*}$, and $U{𝔤𝔩}_n$ the universal enveloping algebra of ${𝔤𝔩}_n$. By Poincaré-Birkhoff-Witt theorem $S({𝔤𝔩}_n)$ is isomorphic to the graded algebra $gr(U{𝔤𝔩}_n)$, associated with the order filtration on $U{𝔤𝔩}_n$. Let $A\subseteq S({𝔤𝔩}_n)$ be a Poisson-commutative subalgebra; one says that a commutative subalgebra $\hat{A}\subseteq U{𝔤𝔩}_n$ is a quantisation of $A$, if its image under the natural projection $U{𝔤𝔩}_n\to gr(U{𝔤𝔩}_n)\cong S({𝔤𝔩}_n)$ is equal to $A$. In my talk I will speak about the so-called "argument shift" subalgebras $A=A_{\xi }$ in $S({𝔤𝔩}_n)$, generated by the iterated derivations of central elements in $S({𝔤𝔩}_n)$ 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 quasi-derivations $\hat{\xi }$ of $U{𝔤𝔩}_n$. These operations are "quantisations" of the derivations on $S({𝔤𝔩}_n)$ and verify an analog of the Leibniz rule. In fact, I will show that iterated quasiderivation of certain generating elements in $U{𝔤𝔩}_n$ are equal to the linear combinations of the elements, earlier constructed by Tarasov.

Video