Ring Theoretic Properties of Quantum Grassmannians

Kelly, Ann C.; Lenagan, T. H.; Rigal, Laurent

doi:10.1142/s0219498804000630

Cited by 36 publications

(84 citation statements)

References 10 publications

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“…Most proofs will be omitted since these results already appear in [5] and [10]. Appropriate references will be given in the text.…”

Section: Basic Definitionsmentioning

confidence: 99%

“…Then, the corresponding property can be transfered to O q (M m,n (k)) using the dehomogenisation map D m,n , introduced in [5], that relates the two algebras. We briefly recall the definition of this map.…”

Section: Basic Definitionsmentioning

confidence: 99%

“…(This is an immediate consequence of the relation [13][24] 4 (k)), see the introduction of [5], using a suitable injection of…”

Section: Quantum Graded Asl and The Maximal Order Propertymentioning

confidence: 99%

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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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“…Most proofs will be omitted since these results already appear in [5] and [10]. Appropriate references will be given in the text.…”

Section: Basic Definitionsmentioning

confidence: 99%

Section: Basic Definitionsmentioning

confidence: 99%

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Quantum Analogues of Schubert Varieties in the Grassmannian

Lenagan¹,

Rigal²

2008

Glasgow Math. J.

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“…By now, many more commutation formulas are known for much larger collections of quantum minors (see [3,6]). The impetus for finding such results has been two-fold: (i) from the point of view of representation theory, such questions are intimately tied to the study of the canonical (or crystal) bases of Lustzig and Kashiwara [1,8,14]; (ii) from the point of view of non-commutative algebraic geometry, the study of quantum determinantal ideals provides non-commutative versions of the classical determinantal varieties [2,7,9]. Our goal is different.…”

Section: [J]][[i]] = Q a [[I]][[j]mentioning

confidence: 99%

QUASI-DETERMINANTS AND q-COMMUTING MINORS

Lauve¹

2010

Glasgow Math. J.

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Abstract. We present two new proofs of the q-commuting property holding among certain pairs of quantum minors of a q-generic matrix. The first uses elementary quasi-determinantal arithmetic; the second involves paths in a directed graph. Together, they indicate a means to build the multi-homogeneous coordinate rings of flag varieties in other non-commutative settings.2010 Mathematics Subject Classification. 20G42, 16T30, 15A15.

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“…The algebra of Example 3.4, whose generators are not indexed by Z, is related to the coordinate ring of quantum n × n-matrices presented for instance in [10].…”

Section: Some Operations On Good Pbw-algebras and Examples Letmentioning

confidence: 99%

A Priddy-type Koszulness criterion for non-locally finite algebras

Brunetti

Ciampella

2007

Colloq. Math.

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Abstract. A celebrated result by S. Priddy states the Koszulness of any locally finite homogeneous PBW-algebra, i.e. a homogeneous graded algebra having a Poincaré-Birkhoff-Witt basis. We find sufficient conditions for a non-locally finite homogeneous PBW-algebra to be Koszul, which allows us to completely determine the cohomology of the universal Steenrod algebra at any prime.Introduction. The notion of Koszul algebra, introduced by S. Priddy in [15] in particular to construct resolutions for the Steenrod algebra, has led to remarkable achievements in the study of associative algebras defined by quadratic relations. The Koszulness condition provides decisive information to solve several basic problems in that context. [14] gives a beautiful and comprehensive account of the impact of Koszul algebras in several areas of mathematics. Such algebras arise in fact in algebraic geometry, representation theory, non-commutative geometry, number theory, and obviously algebraic topology.In this paper we deal with homogeneous algebras A isomorphic to a quotient of the form T (V )/J(R), where T (V ) = i T i is the tensor algebra over a K-vector space V with basis X = {x i | i ∈ I}, I is a (not necessarily bounded) totally ordered set, and J(R) is the two-sided ideal of relations generated by some R ⊂ T 2 = V ⊗ V .Note that all the Koszulness criteria listed for example in [9] concerning the Hilbert series of A become meaningless if I is not finite; even Priddy's criterion, i.e. the existence of a Poincaré-Birkhoff-Witt basis [15], only holds if the algebra has an internal degree induced by a map g : I n → Z and it is locally finite with respect to length and g (see [15]). It follows that there are examples of homogeneous quadratic algebras whose Koszulness cannot be checked using directly the criteria listed in [9] and [14]: Poisson enveloping algebras of Poisson algebras with generators indexed by Z and quadratic brackets (see [11] for the definition), infinite quantum grassmannians (see

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Ring Theoretic Properties of Quantum Grassmannians

Cited by 36 publications

References 10 publications

Quantum Analogues of Schubert Varieties in the Grassmannian

Quantum Analogues of Schubert Varieties in the Grassmannian

QUASI-DETERMINANTS AND q-COMMUTING MINORS

A Priddy-type Koszulness criterion for non-locally finite algebras

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