The argument shift method in universal enveloping algebra $U\mathfrak{gl}_d$

TL;DR

This paper proves a conjecture extending the argument shift method from symmetric algebra to the universal enveloping algebra of l_d, revealing that iterated quasi-derivations of central elements commute, enhancing understanding of argument shift algebras.

Contribution

It establishes that the argument shift procedure can be extended to Ul_d, showing that iterated quasi-derivations of central elements commute, which was previously conjectured.

Findings

Quasi-derivations of central elements commute in Ul_d.

Extension of argument shift method to universal enveloping algebra.

Improved structural understanding of Mishchenko-Fomenko algebras.

Abstract

We prove the conjecture that allows one extend the argument shifting procedure from symmetric algebra $S\mathfrak{gl}_d$ of the Lie algebra $\mathfrak{gl}_d$ to the universal enveloping algebra $U\mathfrak{gl}_d$. Namely, it turns out that the iterated quasi-derivations of the central elements in $U\mathfrak{gl}_d$ commute with each other. Here quasi-derivation is a linear operator on $U\mathfrak{gl}_d$, constructed by Gurevich, Pyatov and Saponov. This allows one better understand the structure of \textit{argument shift algebras} (or \textit{Mishchenko-Fomenko algebras}) in the universal enveloping algebra of $\mathfrak{gl}_d$.