Let Z be an affine Poisson algebra. A Poisson Z-order is a a finite Z-algebra equipped with the structure of a Poisson module. Examples arise in nature when one considers quantum group specialised at a root unity. In this context there are two competing definitions for what we might mean by a Poisson primitive ideal. In this talk I shall present a theorem which states that both of these definitions actually coincide, and I will go on to explain that, under some mild assumptions, every simple Poisson A-module is naturally associated to a symplectic leaf of Spec Z.

 

This work was motivated by a famous conjecture bounding the dimensions of simple modules for quantised enveloping algebras at roots of unity in terms of the dimensions of the leaves.

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