Quasiderivations of the algebra $U\mathfrak{gl}_n$ and the quantum Mischenko–Fomenko algebras

Quasiderivations of the universal enveloping algebra $U\mathfrak{gl}_n$ were first introduced by D. Gurevich, P. Pyatov, and P. Saponov in their study of reflection equation algebras; they are linear operators on $U\mathfrak{gl}_n$ that satisfy certain algebraic relations, which generalise the usual Leibniz rule. In this note, we show that the iterated action of the operator equal to a linear combination of the quasiderivations on a certain set of generators of the center of $U\mathfrak{gl}_n$ (namely on the symmetrised coefficients of the characteristic polynomial) produces commuting elements. The resulting algebra coincides with the quantum Mischenko–Fomenko algebra in $U\mathfrak{gl}_n$, introduced earlier by Tarasov, Rybnikov, Molev, and others.