Proceedings of the Edinburgh Mathematical Society

Table of Contents - October 1999 - Volume , Issue 03  

Research Article

Normalité de certains anneaux déterminantiels quantiques

Laurent Rigala1a2

a1 Université de Poitiers Département de Mathématiques 40, Avenue du Recteur Pineau 86022 Poitiers, France

a2 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

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.

(Received January 05 1998)