Seminar talk, 15 March 2023

From Geometry of Differential Equations
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Speaker: Georgy Sharygin

Title: Quasiderivations and commutative subalgebras of the algebra

Let be the Lie algebra of matrices over a characteristic zero field (one can take or ); let be the Poisson algebra of polynomial functions on , and the universal enveloping algebra of . By Poincaré-Birkhoff-Witt theorem is isomorphic to the graded algebra , associated with the order filtration on . Let be a Poisson-commutative subalgebra; one says that a commutative subalgebra is a quantisation of , if its image under the natural projection is equal to . In my talk I will speak about the so-called "argument shift" subalgebras in , generated by the iterated derivations of central elements in by a constant vector field . There exist several ways to define a quantisation of , 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 using as its generators iterated quasi-derivations of . These operations are "quantisations" of the derivations on and verify an analog of the Leibniz rule. In fact, I will show that iterated quasiderivation of certain generating elements in are equal to the linear combinations of the elements, earlier constructed by Tarasov.