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 $n\times n$ matrices, of Weyl algebras, and of Euclidean and symplectic spaces. The framework for this analysis is based upon certain iterated skew polynomial algebras $A$ 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 $\mathcal {H}$ acting as automorphisms of $A$. The prime spectrum of $A$ is then partitioned into finitely many “$\mathcal {H}$-strata” (two prime ideals lie in the same $\mathcal {H}$-stratum if the intersections of their $\mathcal {H}$-orbits coincide), and we show that a prime ideal $P$ of $A$ is primitive exactly when $P$ is maximal within its $\mathcal {H}$-stratum. This approach relies on a theorem of Moeglin-Rentschler (recently extended to positive characteristic by Vonessen), which provides conditions under which $\mathcal {H}$ acts transitively on the set of rational ideals within each $\mathcal {H}$-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.