The argument shift method in universal enveloping algebra [formula omitted

Abstract

We prove the statement that allows one extend the argument shifting procedure from symmetric algebra Sgld of the Lie algebra gld to the universal enveloping algebra Ugld. Namely, it turns out that the iterated quasi-derivations of the central elements in Ugld commute with each other. Here quasi-derivations are certain linear operators on Ugld, projecting to the partial derivatives on symmetric algebra S(gld). This allows one better understand the structure of argument shift algebras (or Mishchenko-Fomenko algebras) in the universal enveloping algebra of gld.