Seminar talk, 15 March 2023

Speaker: Georgy Sharygin

Title: Quasiderivations and commutative subalgebras of the algebra $U{\mathfrak{𝔤}\mathfrak{𝔩}}_n$

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

Video