Goldie rank of primitive quotients via lattice point enumeration

TL;DR

This paper explores the structure of primitive quotients of certain algebraic rings, showing their Goldie ranks follow a finite set of quasi-polynomials linked to convex geometric Ehrhart theory.

Contribution

It establishes that primitive quotients' Goldie ranks are described by finitely many quasi-polynomials, connecting algebraic properties to convex geometric Ehrhart quasi-polynomials.

Findings

Primitive quotients fall into finitely many families.

Goldie ranks are given by common quasi-polynomials.

Quasi-polynomials are realized as Ehrhart quasi-polynomials.

Abstract

Let k be an algebraically closed field of characteristic 0. Musson and vandenBergh classified primitive ideals for rings of torus invariant differential operators. This classification applies in particular to subquotients of localized extended Weyl algebras where it can be made explicit in terms of convex geometry. We recall these result and then turn to the corresponding primitive quotients and study their Goldie ranks. We prove that the primitive quotients fall into finitely many families whose Goldie ranks are given by a common quasi-polynomial and then realize these quasi-polynomials as Ehrhart quasi-polynomials arising from convex geometry.