The Poisson center and polynomial, maximal Poisson commutative subalgebras, especially for nilpotent Lie algebras of dimension at most seven

TL;DR

This paper studies the Poisson center and maximal Poisson commutative subalgebras of symmetric algebras of nilpotent Lie algebras up to dimension seven, providing explicit descriptions and confirming a conjecture.

Contribution

It offers explicit descriptions of the Poisson center for all indecomposable nilpotent Lie algebras of dimension at most seven and constructs polynomial maximal Poisson commutative subalgebras.

Findings

Explicit Poisson center descriptions for all such Lie algebras.

Construction of polynomial maximal Poisson commutative subalgebras.

Validation of Milovanov's conjecture in this context.

Abstract

Let g be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We collect some general results on the Poisson center of S(g), including some simple criteria regarding its polynomiality, and also on certain Poisson commutative subalgebras of S(g). These facts are then used to finish our earlier work on this subject, i.e. to give an explicit description for the Poisson center of all indecomposable, nilpotent Lie algebras of dimension at most seven. Among other things, we also provide a polynomial, maximal Poisson commutative subalgebra of S(g), enjoying additional properties. As a by-product we show that a conjecture by Milovanov is valid in this situation. These results easily carry over to the enveloping algebra U(g).