The Dixmier-Moeglin equivalence in quantum coordinate rings and quantized Weyl algebras

We study prime and primitive ideals in a unified setting applicable to quantizations (at nonroots of unity) of n x n matrices, of Weyl algebras, and of Euclidean and symplectic spaces. The framework for this analysis is based upon certain iterated skew polynomial algebras A over infinite fields k of arbitrary characteristic. Our main result is the verification, for A, of a characterization of primitivity established by Dixmier and Moeglin for complex enveloping algebras. Namely, we show that a prime ideal P of A is primitive if and only if the center of the Goldie quotient ring of A/P is algebraic over k, if and only if P is a locally closed point - with respect to the Jacobson topology - in the prime spectrum of A. These equivalences are established with the aid of a suitable group H acting as automorphisms of A. The prime spectrum of A is then partitioned into finitely many "H-strata" (two prime ideals lie in the same H-stratum if the intersections of their H-orbits coincide), and we show that a prime ideal P of A is primitive exactly when P is maximal within its H-stratum. This approach relies on a theorem of Moeglin-Rentschler (recently extended to positive characteristic by Vonessen), which provides conditions under which H acts transitively on the set of rational ideals within each H-stratum. In addition, we give detailed descriptions of the strata that can occur in the prime spectrum of A. For quantum coordinate rings of semisimple Lie groups, results analogous to those obtained in this paper already follow from work of Joseph and Hodges-Levasseur-Toro. For quantum affine spaces, analogous results have been obtained in previous work of the authors.