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

Abstract

We study prime and primitive ideals in a unified setting applicable to quantizations (at nonroots of unity) of n × n n\times 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 A over infinite fields k k of arbitrary characteristic. Our main result is the verification, for A A , of a characterization of primitivity established by Dixmier and Moeglin for complex enveloping algebras. Namely, we show that a prime ideal P P of A A is primitive if and only if the center of the Goldie quotient ring of A / P A/P is algebraic over k k , if and only if P P is a locally closed point – with respect to the Jacobson topology – in the prime spectrum of A A . These equivalences are established with the aid of a suitable group H \mathcal {H} acting as automorphisms of A A . The prime spectrum of A A is then partitioned into finitely many “ H \mathcal {H} -strata” (two prime ideals lie in the same H \mathcal {H} -stratum if the intersections of their H \mathcal {H} -orbits coincide), and we show that a prime ideal P P of A A is primitive exactly when P P is maximal within its H \mathcal {H} -stratum. This approach relies on a theorem of Moeglin-Rentschler (recently extended to positive characteristic by Vonessen), which provides conditions under which H \mathcal {H} acts transitively on the set of rational ideals within each H \mathcal {H} -stratum. In addition, we give detailed descriptions of the strata that can occur in the prime spectrum of A 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.