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 Poincaré-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.

mathematical physicsanalysis of PDEsdifferential geometry

Audience: researchers in the topic

( video )

Geometry of differential equations seminar