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Prime ideals of quantized Weyl algebras. (English) Zbl 0881.16012
The algebras of the title, denoted $$A_n^{{\overline q},\Lambda}$$ (where $$\overline q$$ is an $$n$$-vector and $$\Lambda$$ a multiplicatively antisymmetric $$n\times n$$ matrix of nonzero scalars), were introduced by E. E. Demidov [Usp. Mat. Nauk 48, No. 6, 39-74 (1993); English transl.: Russ. Math. Surv. 48, No. 6, 41-79 (1993; Zbl 0839.17011)], G. Maltsiniotis [Calcul différentiel quantique, Groupe de travail, Université Paris VII (1992)], and others. Here, the authors compute the prime spectrum of $$A_n^{{\overline q},\Lambda}$$, under the assumption that certain subgroups of the multiplicative group generated by the entries of $$\overline q$$ and $$\Lambda$$ have maximal rank. In particular, the prime ideals of $$A_n^{{\overline q},\Lambda}$$ are all polynormal, there are infinitely many maximal ideals (all of height $$2n$$), while there are only finitely many nonmaximal prime ideals. (Similar results were obtained, using different methods, by L. Rigal [Beitr. Algebra Geom. 37, No. 1, 119-148 (1996; Zbl 0876.17012)].) The authors also investigate a related algebra $${\mathcal A}_n^{{\overline q},\Lambda}$$, which shares with $$A_n^{{\overline q},\Lambda}$$ the simple localization $$B_n^{{\overline q},\Lambda}$$ studied by the second author [J. Algebra 174, No. 1, 267-281 (1995; Zbl 0833.16025)]. In this algebra, the prime ideals are again polynormal, but there are only finitely many of them if $$n>1$$.
A different description of $$\text{spec }A_n^{{\overline q},\Lambda}$$ is implicit in work of T. H. Lenagan and the reviewer [J. Pure Appl. Math. 111, 1-3, 123-142 (1996; Zbl 0864.16018)], and is given explicitly in work of E. S. Letzter and the reviewer [The Dixmier-Moeglin equivalence in quantum matrices and quantized Weyl algebras (to appear)]. In these papers, the only restriction on the parameters is that no entry of $$\overline q$$ is a root of unity.

##### MSC:
 16P40 Noetherian rings and modules (associative rings and algebras) 16D25 Ideals in associative algebras 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16S36 Ordinary and skew polynomial rings and semigroup rings
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