On the semi-centre of a Poisson algebra

TL;DR

This paper generalizes the concept of the semi-centre from Lie algebras to integral Poisson algebras, establishing conditions for its structure and properties, including grading and the behavior of Casimirs.

Contribution

It introduces a generalized framework for the Poisson semi-centre in integral Poisson algebras and characterizes when it forms a graded Poisson algebra with desirable properties.

Findings

Necessary and sufficient conditions for Poisson semi-centre to be graded

The semi-centre's rational Casimirs are quotients of Poisson normal elements

Poisson Dixmier-M{\

Abstract

If $\mathfrak{g}$ is a Lie algebra then the semi-centre of the Poisson algebra $S(\mathfrak{g})$ is the subalgebra generated by ad$(\mathfrak{g})$-eigenvectors. In this paper we abstract this definition to the context of integral Poisson algebras. We identify necessary and sufficient conditions for the Poisson semi-centre $A^{\operatorname{sc}}$ to be a Poisson algebra graded by its weight spaces. In that situation we show the Poisson semi-centre exhibits many nice properties: the rational Casimirs are quotients of Poisson normal elements and the Poisson Dixmier-M{\oe}glin equivalence holds for the semi-centre.