Complete commutative subalgebras in polynomial Poisson algebras: a proof of the Mischenko--Fomenko conjecture

TL;DR

This paper proves the Mischenko-Fomenko conjecture, demonstrating the existence of complete sets of commuting polynomials on the dual of any finite-dimensional Lie algebra, thus establishing polynomial integrability in Lie-Poisson systems.

Contribution

It provides an explicit geometric construction for commuting polynomials on the dual space of Lie algebras, extending Sadetov's proof with concrete examples.

Findings

Confirmed the existence of complete commutative polynomial subalgebras in polynomial Poisson algebras.

Developed an explicit geometric method for constructing commuting polynomials.

Illustrated the construction with specific examples.

Abstract

The Mishchenko-Fomenko conjecture says that for each real or complex finite-dimensional Lie algebra $\goth g$ there exists a complete set of commuting polynomials on its dual space $\goth g^*$. In terms of the theory of integrable Hamiltonian systems this means that the dual space $\goth g^*$ endowed with the standard Lie-Poisson bracket admits polynomial integrable Hamiltonian systems. Recently this conjecture has been proved by S.T. Sadetov. Following his idea, we give an explicit geometric construction for commuting polynomials on $\goth g^*$ and consider some examples.