Seminar talk, 6 March 2024

Speaker: Georgy Sharygin

Title: Deformation quantisation of the argument shift on $U{\mathfrak {gl}}(n)$

Abstract:
Argument shift algebras are the commutative subalgebras in the symmetric algebras of a Lie algebra, generated by the iterated derivations (in direction of a constant vector field) of Casimir elements in $S{\mathfrak {gl}}(n)$. In particular all these quasiderivations do mutually commute. In my talk I will show that a similar statement holds for the algebra $U{\mathfrak {gl}}(n)$ and its quasiderivations: namely, I will show that iterated quasiderivations of the central elements of $U{\mathfrak {gl}}(n)$ with respect to a constant quasiderivation do mutually commute. Our proof is based on the existence and properties of "Quantum Mischenko-Fomenko" algebras, and (which is worse) cannot be extended to other Lie algebras, but we believe that the fact that the "shift operator" can be raised to $U{\mathfrak {gl}}(n)$ is an interesting fact.