On the Semi-centre of a Poisson Algebra

If g is a Lie algebra then the semi-centre of the Poisson algebra S(g) is the subalgebra generated by ad(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 Asc 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–Mœglin equivalence holds for Asc.