Prime ideals of quantized Weyl algebras

Akhavizadegan, M. ; Jordan, D. A.

Glasgow mathematical journal, Tome 38 (1996) no. 3, pp. 283-297

Résumé

The main object of study in this paper is the quantized Weyl algebra which arises from the work of Maltsiniotis [10] on noncommutative differential calculus. This algebra has been studied from the point of view of noncommutative ring theory by various authors including Alev and Dumas [1], the second author [9], Cauchon [3], and Goodearl and Lenagan [5]. In [9], it is shown that has n normal elements zi and, subject to a condition on the parameters, the localization obtained on inverting these elements is simple of Krull and global dimension n. It is easy to show that each of these normal elements generates a height one prime ideal and that these are all the height one prime ideals of . The purpose of this paper is to determine, under a stronger condition on the parameters, all the prime ideals of and to compare the prime spectrum with that of a related algebra . This algebra has more symmetric defining relations than those of but it shares the same simple localization which again is obtained by inverting n normal elements zi. Like the alternative algebra can be regarded as an algebra of skew differential (or difference) operators on the coordinate ring of quantum n-space.

Détail

DOI : 10.1017/S0017089500031712

Bibliographie

[1] 1.Alev, J. and Dumas, F., Sur le corps des fractions de certaines algèbres quantiques, J. Algebra 170 (1994), 229–265. Google Scholar | DOI

[2] 2.Bryant, V. W., Aspects of combinatorics (Cambridge University Press, 1993). Google Scholar

[3] 3.Cauchon, G., Quotient premiers de O (M (k)), preprint, Université de Reims. Google Scholar

[4] 4.Goodearl, K. R., Prime ideals in skew polynomial rings and quantized Weyl algebras, J. Algebra 150 (1992), 324–377. Google Scholar | DOI

[5] 5.Goodearl, K. R. and Lenagan, T. H., Catenarity in quantum algebras, preprint, Universities of California (Santa Barbara) and Edinburgh. Google Scholar | DOI

[6] 6.Goodearl, K. R. and Letzter, E. S., Prime ideals in skew and q-skew polynomial rings, Mem. Amer. Math. Soc. 109 (1994), no. 521. Google Scholar

[7] 7.Goodearl, K. R. and Warfield, R. B. Jr, An introduction to noncommutative Noetherian rings (Cambridge University Press, 1989). Google Scholar

[8] 8.Jordan, D. A., Normal elements and completions of non-commutative Noetherian rings, Bull. London Math. Soc. 19 (1987), 417–424. Google Scholar | DOI

[9] 9.Jordan, D. A., A simple localization of the quantized Weyl algebra, J. Algebra 174 (1995), 267–281. Google Scholar | DOI

[10] 10.Maltsiniotis, G., Groupes quantique et structures diffdrentielles, C.R. Acad. Sci. Paris Sir. I Math. 311 (1990), 831–834. Google Scholar

[11] 11.McConnell, J. C. and Pettit, J. J., Crossed products and multiplicative analogues of Weyl algebras, J. London Math. Soc. (2) 38 (1988), 47–55. Google Scholar | DOI

[12] 12.McConnell, J. C. and Robson, J. C., Noncommutative Noetherian rings (Wiley, 1987). Google Scholar

[13] 13.Oh, S.Q., Primitive ideals of the coordinate ring of quantum symplectic space, J. Algebra 174 (1995), 531–552. Google Scholar | DOI

[14] 14.Rigal, L., Spectre de l'algèbre de Weyl quantique, preprint, Université de Paris VI (1994). Google Scholar