The Maximal Order Property for Quantum Determinantal Rings

Lenagan, T. H.; Rigal, Laurent

doi:10.1017/s0013091502000809

Cited by 7 publications

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“…These relations agree with the relations used in [3], [5], [6], [10] and [11], but they differ from those of [12] by an interchange of q and q −1 . It is well known that the ring R is a Noetherian domain.…”

Section: H-invariant Prime Ideals In O Q (M Mp (C))supporting

confidence: 83%

“…Using this new tool, we can prove that, if u 3, the ideal in O q (M m,p (C)) generated by Y 1,p and the u × u quantum minors is prime. This result allows Lenagan and Rigal [11] to show that the quantum determinantal factor rings of O q (M m,p (C)) are maximal orders.…”

Section: Introductionmentioning

confidence: 76%

“…We deduce from the above results that J w is generated by Remark 10.7. Corollary 10.6 allowed Lenagan and Rigal [11] to show that the quantum determinantal factor rings of O q (M m,p (C)) are maximal orders.…”

Section: A Criterion For 1 × 1 Quantum Minorsmentioning

confidence: 99%

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GENERATORS FOR $\mathcal{H}$-INVARIANT PRIME IDEALS IN $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$

Launois¹

2004

Proceedings of the Edinburgh Mathematical Society

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It is known that, for generic q, the H-invariant prime ideals in Oq (Mm,p(C)) are generated by quantum minors (see S. Launois, Les idéaux premiers invariants de Oq (Mm,p(C)), J. Alg., in press). In this paper, m and p being given, we construct an algorithm which computes a generating set of quantum minors for each H-invariant prime ideal in Oq (Mm,p(C)). We also describe, in the general case, an explicit generating set of quantum minors for some particular H-invariant prime ideals in Oq (Mm,p(C)). In particular, if (Y i,α ) (i,α)∈[ [1,m] ]×[ [1,p] ] denotes the matrix of the canonical generators of Oq(Mm,p(C)),we prove that, if u 3, the ideal in Oq (Mm,p(C)) generated by Y 1,p and the u × u quantum minors is prime. This result allows Lenagan and Rigal to show that the quantum determinantal factor rings of Oq (Mm,p(C)) are maximal orders (see T. H. Lenagan and L. Rigal, Proc. Edinb. Math. Soc. 46 (2003), 513-529).Keywords: quantum matrices; quantum minors; prime ideals; quantum determinantal ideals; deleting-derivations algorithms 2000 Mathematics subject classification: Primary 16P40 Secondary 16W35; 20G42 IntroductionFix two positive integers m and p with m, p 2 and consider some complex number q which is transcendental over Q. Denote by R = O q (M m,p (C)) the quantization of the ring of regular functions on m × p matrices with entries in C (the field of complex numbers) and let (Y i,α (If m = p, this action is induced by the bialgebra structure of R and, if m = p, it is easy to check that the relations which define R are preserved by the group H.) It is known from work of Goodearl and Letzter that R has only finitely many Hinvariant prime ideals (see [8]) and that, in order to calculate the prime and primitive spectra of R, it is enough to determine the H-invariant prime ideals of R (see [8, Theorem 6.6]).In [10], we proved that the H-invariant prime ideals in R are generated by quantum minors, as conjectured by Goodearl and Lenagan (see [5] and [6]). In this paper, we use 163 164 S. Launois this result, together with Cauchon's description for the set of H-invariant prime ideals of R (see [3, Théorème 3.2.1]), to construct an algorithm which provides an explicit generating set of quantum minors for each H-invariant prime ideal in R (see § 4). (Of course, these generating sets can be computed with this algorithm only when m and p have fixed values.)The last part of this paper is devoted to the general case. We construct certain sets of quantum minors which generate prime ideals of R. In order to do that, we consider a new deleting-derivations algorithm (see [2]) that we define in § 5. Using this new tool, we can prove that, if u 3, the ideal in O q (M m,p (C)) generated by Y 1,p and the u × u quantum minors is prime. This result allows Lenagan and Rigal [11] to show that the quantum determinantal factor rings of O q (M m,p (C)) are maximal orders. H-invariant prime ideals in O q (M m,p (C))Throughout this paper, we use the following conventions.(i) N, Q and C denote, respectively, the set of natur...

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Section: H-invariant Prime Ideals In O Q (M Mp (C))supporting

confidence: 83%

Section: Introductionmentioning

confidence: 76%

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GENERATORS FOR $\mathcal{H}$-INVARIANT PRIME IDEALS IN $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$

Launois¹

2004

Proceedings of the Edinburgh Mathematical Society

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“…Hence, as a consequence of our results, we are able to show that quantum determinantal rings are maximal orders in their division ring of fractions. This generalises results obtained in [9] where it was shown that quantum determinantal rings are maximal orders under the restrictive hypotheses that k is the field of complex numbers and the deformation parameter q is transcendental over ‫.ޑ‬…”

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confidence: 86%

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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“…The algebras that we study are noncommutative analogues of coordinate rings of natural varieties arising from Lie theory and we want to study them as such. This was already the point of view in the works [6,7] and [8], where properties of geometric nature of related algebras, expressible either in ring theoretic language (integrity, normality), or homologically (AS-Cohen-Macaulay, AS-Gorenstein properties) were studied.…”

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confidence: 97%

Generalised Quantum Determinantal Rings Are Maximal Orders

Lenagan

Rigal

2020

Glasgow Math. J.

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Generalised quantum determinantal rings are the analogue in quantum matrices of Schubert varieties. Maximal orders are the noncommutative version of integrally closed rings. In this paper, we show that generalised quantum determinantal rings are maximal orders. The cornerstone of the proof is a description of generalised quantum determinantal rings, up to a localisation, as skew polynomial extensions.

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The Maximal Order Property for Quantum Determinantal Rings

Abstract: We develop a method of reducing the size of quantum minors in the algebra of quantum matrices Oq(Mn). We use the method to show that the quantum determinantal factor rings of Oq(Mn(C)) are maximal orders, for q an element of C transcendental over Q.

Cited by 7 publications

References 10 publications

GENERATORS FOR $\mathcal{H}$-INVARIANT PRIME IDEALS IN $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$

GENERATORS FOR $\mathcal{H}$-INVARIANT PRIME IDEALS IN $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$

Quantum Analogues of Schubert Varieties in the Grassmannian

Generalised Quantum Determinantal Rings Are Maximal Orders

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