zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] Goodearl, Prime ideals in skew and q-skew polynomial rings, Mem. Amer. Math. Soc. 109 pp 521– (1994) · Zbl 0814.16026
[2] Goodearl, Catenarity in quantum algebras · Zbl 0864.16018 · doi:10.1016/0022-4049(95)00120-4
[3] DOI: 10.1016/S0021-8693(05)80036-5 · Zbl 0779.16010 · doi:10.1016/S0021-8693(05)80036-5
[4] Cauchon, Quotient premiers de
[5] Bryant, Aspects of combinatorics (1993) · Zbl 0777.05001
[6] DOI: 10.1006/jabr.1994.1336 · Zbl 0820.17015 · doi:10.1006/jabr.1994.1336
[7] Goodearl, An introduction to noncommutative Noetherian rings (1989) · Zbl 0679.16001
[8] DOI: 10.1006/jabr.1995.1138 · Zbl 0828.17011 · doi:10.1006/jabr.1995.1138
[9] McConnell, Noncommutative Noetherian rings (1987) · Zbl 0644.16008
[10] DOI: 10.1112/jlms/s2-38.1.47 · Zbl 0652.16007 · doi:10.1112/jlms/s2-38.1.47
[11] DOI: 10.1006/jabr.1995.1128 · Zbl 0833.16025 · doi:10.1006/jabr.1995.1128
[12] Maltsiniotis, C.R. Acad. Sci. Paris Sir. I Math. 311 pp 831– (1990)
[13] DOI: 10.1112/blms/19.5.417 · Zbl 0627.16013 · doi:10.1112/blms/19.5.417
[14] Rigal, Spectre de l’algèbre de Weyl quantique (1994) · Zbl 0876.17012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.