Seminar talk, 6 March 2024

From Geometry of Differential Equations
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Speaker: Georgy Sharygin

Title: Deformation quantisation of the argument shift on

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 . In particular all these quasiderivations do mutually commute. In my talk I will show that a similar statement holds for the algebra and its quasiderivations: namely, I will show that iterated quasiderivations of the central elements of 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 is an interesting fact.