On the quantum argument shift method

In a recent work by two of us the argument shift method was extended from the symmetric algebra $\mathrm{S}(\mathfrak{g})$ of the general linear Lie algebra $\mathfrak{g}$ to the universal enveloping algebra $\mathrm{U}(\mathfrak{g})$. We show in this paper that some features of this ‘quantum argument shift method’ can be applied to the remaining classical matrix Lie algebras $\mathfrak{g}$. We prove that a single application of the quasi-derivation to central elements of $\mathrm{U}(\mathfrak{g})$ yields elements of the corresponding quantum Mishchenko-Fomenko subalgebra. We show that generators of this subalgebra can be obtained by iterated application of the quasi-derivation to generators of the center of $\mathrm{U}(\mathfrak{g})$.