On the quantum argument shift method

TL;DR

This paper extends the quantum argument shift method to classical matrix Lie algebras, demonstrating how to generate quantum Mishchenko-Fomenko subalgebras via quasi-derivations applied to central elements.

Contribution

It applies the quantum argument shift method to all classical matrix Lie algebras, providing a systematic way to generate quantum Mishchenko-Fomenko subalgebras.

Findings

Single quasi-derivation yields elements of the quantum Mishchenko-Fomenko subalgebra.

Generators of the subalgebra can be obtained by iterated quasi-derivations.

Method applies to all classical matrix Lie algebras.

Abstract

In a recent work by two of us the argument shift method was extended from the symmetric algebra ${\rm S}({\mathfrak g})$ of the general linear Lie algebra ${\mathfrak g}$ to the universal enveloping algebra ${\rm U}({\mathfrak g})$. 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 ${\rm U}({\mathfrak g})$ 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 ${\rm U}({\mathfrak g})$.