Quasidifferential operator and quantum argument shift method

We describe an explicit formula for the first-order quasiderivation of an arbitrary central element of the universal enveloping algebra of a general linear Lie algebra. We apply it to show that derivations of any two central elements of the universal enveloping algebra commute. This contributes to the Vinberg problem of finding commutative subalgebras in universal enveloping algebras with the underlying Poisson algebras determined by the argument shift method.

Automated summary

In this study, we provide an explicit formula for the first-order quasiderivation of an arbitrary central element in the universal enveloping algebra. We also show that derivations of any two central elements in the universal enveloping algebra are commutative. This contributes to solving the Vinberg problem of finding commutative subalgebras in universal enveloping algebras with the underlying Poisson algebras determined by the argument shift method.