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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: Laurent.Rigal@univ-st-etienne.fr
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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

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