The quantum argument shift method on $U\mathfrak{gl}_n$

The term "argument shift method" is used for a simple and efficient method to construct commutative subalgebras in Poisson algebras by deforming the Casimir elements in them. This method is primarily used to search for Poisson commutative subalgebras in symmetric algebras of various Lie algebras; it is closely related with the bi-Hamiltonian induction (Lenard-Magri scheme). However little is known about the possible extension of this method to the quantum algebras associated with given Poisson algebras; this is true even for the symmetric algebra of a Lie algebra, where the quantization is well known (it is equal to the universal enveloping algebra). I will tell about a particular case, the algebra $U\mathfrak{gl}_n$, for which one can find a shifting operator raising to this algebra the shift on $S(\mathfrak{gl}_n)$, and prove that this operator verifies the same condition as before: when used to deform the elements in the center of $U\mathfrak{gl}_n$, it yields a set of commuting elements.
The talk is partially based on joint works with Y.Ikeda.