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Normalité de certains anneaux déterminantiels quantiques

Published online by Cambridge University Press:  20 January 2009

Laurent Rigal
Laurent Rigal
Affiliation:
Université de Poitiers Département de Mathématiques 40, Avenue du Recteur Pineau 86022 Poitiers, France Current address: Université de Saint-Etienne Faculté des Sciences et Techniques, Mathématiques 23, Rue du Docteur Paul Michelon, 42023 Saint-Etienne Cedex, France, E-mail address: [email protected]
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Let Kq[X] = Oq(M(m, n)) be the quantization of the ring of regular functions on m × n matrices and Iq (X) be the ideal generated by the 2 × 2 quantum minors of the matrix X=(Xij)l≤i≤m, I≤j≤n of generators of Kq[X]. The residue class ring Rq(X) = Kq[X]/Iq(X) (a quantum analogue of determinantal rings) is shown to be an integral domain and a maximal order in its divisionring of fractions. For the proof we use a general lemma concerning maximalorders that we first establish. This lemma actually applies widely to prime factors of quantum algebras. We also prove that, if the parameter isnot a root of unity, all the prime factors of the uniparameter quantum space are maximal orders in their division ring of fractions.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 42 , Issue 3 , October 1999 , pp. 621 - 640
Copyright
Copyright © Edinburgh Mathematical Society 1999

References

RÉFÉRENCES

1.Brown, K. A. and Goodearl, K. R., Prime spectra of quantum semisimple groups, Trans. Amer. Math. Soc. 348 (1996), 2465–2502.CrossRefGoogle Scholar
2.Bruns, W. and Vetter, U., Determinantal rings (Lecture Notes in Math. 1327, Springer-Verlag, Berlin, 1988).CrossRefGoogle Scholar
3.Chamarie, M., Localisations dans les ordres maximaux, Comm. Algebra 2 (1974), 279–293.CrossRefGoogle Scholar
4.Chamarte, M., Sur les ordres maximaux au sens d'Asano (Vorlesungen aus dem Fachbereich Mathematik der Universität Essen, Heft 3, 1979).Google Scholar
5.Chatters, A. W. and Jordan, D. A., Noncommutative unique factorisation rings, J. London Math. Soc. (2) 33 (1986), 22–32.CrossRefGoogle Scholar
6.Goodearl, K. R., Prime ideals in skew polynomial rings and quantized Weyl algebras, J. Algebra 150 (1992), 324–377.CrossRefGoogle Scholar
7.Goodearl, K. R. and Lenagan, T. H., Quantum determinantal ideals, preprint.Google Scholar
8.Maury, G., Ordres maximaux preque-factoriels, Comm. Algebra 8 (1980), 1711–1720.CrossRefGoogle Scholar
9.Maury, G. et Raynaud, J., Ordres maximaux au sens de K. Asano (Lecture Notes in Math. 808, Springer-Verlag, Berlin, 1980).CrossRefGoogle Scholar
10.McConnell, J. C. and Robson, J. C., Noncommutative Noetherian Rings (Wiley, Chichester, 1987).Google Scholar
11.Noumi, M., Yamada, H. and Mimachi, K., Finite dimensional representationsof the quantum group GL q(n; C) and the zonal spherical functions on U q(n – 1)\U q(n), Japan. J. Math. 19 (1993), 31–80.CrossRefGoogle Scholar
12.Parshall, B. and Wang, J., Quantum Linear Groups, Mem. Amer. Math. Soc. 89 (1991), no. 439.Google Scholar
13.Rigal, L., Spectre de l'algèbre de Weyl quantique, Beiträge Algebra Geom. 37 (1996), 119–148.Google Scholar
14.Rigal, L., Analogues quantiques de l'Algèbre de Weyl et Ordres Maximaux quantiques (Thèse de Doctorat de l'Université Paris VI, 1996).Google Scholar
15.Sharpe, D. W., On certain polynomial ideals defined by matrices, Quart. J. Math. Oxford Ser. (2) 15 (1964), 155–175.CrossRefGoogle Scholar
16.Smith, S. P., Quantum groups: an introduction and survey for ring theorists, in Noncommutative Rings (Berkeley, CA, 1989), (Math. Sci. Res. Inst. Publ. 24, Springer-Verlag, New York, 1992), 131–178.CrossRefGoogle Scholar