Centralizers of certain quadratic elements in Poisson--Lie algebras and Argument Shift method

TL;DR

This paper investigates the structure of maximal Poisson-commutative subalgebras in semisimple Lie algebra Poisson algebras, proving their relation to quadratic elements and establishing the uniqueness of their quantization.

Contribution

It demonstrates that Mischenko-Fomenko subalgebras are Poisson centralizers of quadratic elements and proves the uniqueness of their quantization in the universal enveloping algebra.

Findings

Mischenko-Fomenko subalgebras are Poisson centralizers of quadratic elements.

There is a unique quantization of these subalgebras.

The quantization lifts are unique in the universal enveloping algebra.

Abstract

We study maximal Poisson-commutative subalgebras in the Poisson algebra $S(\mathfrak{g})$ of a semisimple Lie algebra $\mathfrak{g}$ constructed by Mischenko and Fomenko with the help of the argument shift method. We prove that these subalgebras are Poisson centralizers of certain quadratic elements of $S(\mathfrak{g})$. We deduce from this that there is a unique quantization of Mischenko--Fomenko subalgebras, i.e. there is a unique way to lift Mischenko--Fomenko subalgebras to commutative subalgebras of the universal enveloping algebra $U(\mathfrak{g})$.