Normalité de certains anneaux déterminantiels quantiques

Rigal, Laurent

doi:10.1017/s0013091500020563

Cited by 4 publications

(4 citation statements)

References 10 publications

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“…Here, we are interested in the question of whether or not R t (X) is a maximal order in its division ring of fractions. This has already been established for R 2 (X) in [10]. We will show that the localized ring R t (X)[x…”

Section: Introductionsupporting

confidence: 69%

“…Also, recall that R 2 (X) has been shown to be a maximal order in [10]. In this section, we prove that R t (X) is a maximal order in its division ring of fractions, when K = C and for q an element of C transcendental over Q.…”

Section: Quantum Determinantal Rings Are Maximal Ordersmentioning

confidence: 80%

“…The following lemma from [10] (see [10,Lemma 1.1]) is recalled here for the convenience of the reader. Let τ be the automorphism of R associated with x; that is ax = xτ (a) for all a ∈ R.…”

Section: Quantum Determinantal Rings Are Maximal Ordersmentioning

confidence: 99%

“…In this case, the ideal det q , X 12 is semiprime; in fact, it is the intersection of two completely prime ideals, each of which is fixed by the automorphism determined by X 1n , so Lemma 4.1 is applicable. However, this case has already been dealt with in [10].…”

Section: O Q (M N )/ Det Q Is a Maximal Ordermentioning

confidence: 99%

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The Maximal Order Property for Quantum Determinantal Rings

Lenagan

Rigal

2003

Proceedings of the Edinburgh Mathematical Society

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Section: Introductionsupporting

confidence: 69%

Section: Quantum Determinantal Rings Are Maximal Ordersmentioning

confidence: 80%

Section: Quantum Determinantal Rings Are Maximal Ordersmentioning

confidence: 99%

Section: O Q (M N )/ Det Q Is a Maximal Ordermentioning

confidence: 99%

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The Maximal Order Property for Quantum Determinantal Rings

Lenagan

Rigal

2003

Proceedings of the Edinburgh Mathematical Society

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Quantized Rank R Matrices

Jakobsen¹,

Jøndrup²

2001

Journal of Algebra

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First some old as well as new results about P.I. algebras, Ore extensions, and degrees are presented. Then quantized n = r matrices as well as certain quantized rq 1 Ž . Ž . rq 1 Ž . factor algebras M n of M n are analyzed. For r s 1, . . . , n y 1, M n is q q q the quantized function algebra of rank r matrices obtained by working modulo the Ž . Ž . ideal generated by all r q 1 = r q 1 quantum subdeterminants and a certain localization of this algebra is proved to be isomorphic to a more manageable one. In almost all cases, the quantum parameter is a primitive mth root of unity. The degrees and centers of the algebras are determined when m is a prime and the general structure is obtained for arbitrary m. ᮊ 2001 Elsevier Science

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Quantum Analogues of Schubert Varieties in the Grassmannian

Lenagan¹,

Rigal²

2008

Glasgow Math. J.

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Abstract. We study quantum Schubert varieties from the point of view of regularity conditions. More precisely, we show that these rings are domains that are maximal orders and are AS-Cohen-Macaulay and we determine which of them are AS-Gorenstein. One key fact that enables us to prove these results is that quantum Schubert varieties are quantum graded algebras with a straightening law that have a unique minimal element in the defining poset. We prove a general result showing when such quantum graded algebras are maximal orders. Finally, we exploit these results to show that quantum determinantal rings are maximal orders.2000 Mathematics Subject Classification. 16W35; 16P40; 16S38; 17B37; 20G42.Introduction. Since the appearance of quantum groups in the eighties, there have been several attempts to define quantum analogues of coordinate rings of grassmannian varieties and, more generally, of flag varieties. Here, we are interested in such deformations for grassmannian varieties and we follow the approach of Lakshmibai and Reshetikhin (see [8]). Hence, we start with the (usual) quantum deformation, denoted by O q (M m,n (k)), of the coordinate ring of m × n matrices. Then, denoting by G m,n (k) the grassmanian of m-dimensional subspaces in k n , the quantum deformation of the coordinate ring of G m,n (k) that we consider is the k-subalgebra of O q (M m,n (k)) generated by the maximal quantum minors. We denote this algebra by O q (G m,n (k)) and call it the quantum grassmannian for simplicity. Precise definitions are recalled in Section 1.Our main interest, in this paper, is the study of a family of quotients of O q (G m,n (k)) that appear in [8] and are natural quantum analogues of coordinate rings of Schubert varieties in G m,n (k). These quantum Schubert varieties have already been studied, to some extent, in [10]. There, they were used as a central tool to show that O q (G m,n (k)) is a quantum graded algebra with a straightening law. Details on the notion of quantum graded algebra with a straightening law (quantum graded A.S.L. for short) can be

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Normalité de certains anneaux déterminantiels quantiques

Abstract: (M{m, n)) be the quantization of the ring of regular functions on m x n matrices and /,(X) be the ideal generated by the 2 x 2 quantum minors of the matrix X = (X,j) l Show more

Cited by 4 publications

References 10 publications

The Maximal Order Property for Quantum Determinantal Rings

The Maximal Order Property for Quantum Determinantal Rings

Quantized Rank R Matrices

Quantum Analogues of Schubert Varieties in the Grassmannian

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