Quantum linear groups

Parshall, Brian; Wang, Jian Pan

doi:10.1090/memo/0439

Cited by 225 publications

(295 citation statements)

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“…(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%

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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“…(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%

Quantum Analogues of Schubert Varieties in the Grassmannian

Lenagan¹,

Rigal²

2008

Glasgow Math. J.

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“…This fact follows immediately from the connection described above with the representation theory of GL n ,q(k) and the fact that the weights in any irreducible GL n ,q (k) -module are stable under the Weyl group ©" [PW;(8.8.2)]. In fact, L q ((l n )) corresponds precisely to the one-dimensional module on GL n ,q (k) defined by the quantum determinant det g .…”

Section: The Category Of 0-torsion -Free B S L S (Q)~modules To the mentioning

confidence: 85%

Generic and $q$-Rational Representation Theory

Cline¹,

Parshall²,

Scott³

1999

Publ. Res. Inst. Math. Sci.

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Part I of this paper develops various general concepts in generic representation and cohomology theories. Roughly speaking, we provide a general theory of orders in non-semisimple algebras applicable to problems in the representation theory of finite and algebraic groups, and we formalize the notion of a "generic" property in representation theory. Part II makes new contributions to the non-describing representation theory of finite general linear groups. First, we present an explicipt Morita equivalence connecting GL n (q] with the theory of g-Schur algebras, extending a unipotent block equivalence of Takeuchi [T]. Second, we apply this Morita equivalence to study the cohomology groups H m (GL n (q),L), when L is an irreducible module in non-describing characteristic.The generic theory of Part I then yields stability results for various groups H l (GL n (q}, L), reminscent of our general theory [CPSK] with van der Kallen of generic cohomology in the describing characteristic case, (in turn, the stable value of such a cohomology group can be expressed in terms of the cohomology of the affine Lie algebra gln(c).) The arguments entail new applications of the theory of tilting modules for #~Schur algebras. In particular, we obtain new complexes involving tilting modules associated to endomorphism algebras obtained from general finite Coxeter groups.

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“…The same construction works if one replaces g by b, p J or l J . This construction is dual to the construction given in [PW,Chapter 9] and analogous to the graded G 1 T -category for reductive algebraic group schemes G in positive characteristic. For a fixed l > 1, let W l = W lZΦ, a subgroup of the affine Weyl group, and W l = W lX, a subgroup of the extended affine Weyl group.…”

Section: ])mentioning

confidence: 99%

An analog of Kostant’s theorem for the cohomology of quantum groups

2010

Proc. Amer. Math. Soc.

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Abstract. We prove the analog of Kostant's Theorem on Lie algebra cohomology in the context of quantum groups. In particular, it is shown that Kostant's cohomology formula holds for quantum groups at a generic parameter q, recovering an earlier result of Malikov in the case where the underlying semisimple Lie algebra g = sl(n). We also show that Kostant's formula holds when q is specialized to an -th root of unity for odd ≥ h − 1 (where h is the Coxeter number of g) when the highest weight of the coefficient module lies in the lowest alcove. This can be regarded as an analog of results of Friedlander-Parshall and Polo-Tilouine on the cohomology of Lie algebras of reductive algebraic groups in prime characteristic.1. Introduction 1.1. Let g be a complex semisimple Lie algebra, L(λ) be a finite dimensional gmodule, p J be a parabolic subalgebra, u J be the nilradical of p J and l J be the associated Levi subalgebra. A celebrated result of Kostant states that the cohomology H• (u J , L(λ)) is the (multiplicity-free) direct sum of finite dimensional l J -modules, L J (w · λ), with w ∈ J W (minimal length coset representatives of W J \W where W J is the Weyl group associated to the subset J of simple roots). There are many proofs of this result (e.g. [Kna, GW]), including ones which involve constructing BGG resolutions [RC, HeK]. Recently the authors [UGA] provided a proof of Kostant's theorem by utilizing linkage in the parabolic category O J .In this paper we will prove an analog of Kostant's theorem in the case of quantum groups over generic parameters and at roots of unity. Our proof uses many of the ideas presented in [UGA]. More specifically, we first consider a subalgebra U q (u J ) of the quantized enveloping algebra U q (g) which is explicitly defined in [BNPP, §2.4, §2.6] using deep results of Lusztig on the existence of a PBW basis. In the classical case, the universal enveloping algebra of u J is a Hopf algebra and the cohomology of this algebra can be computed by using a complex which determines the (ordinary) Lie algebra cohomology. However, in the quantum case the subalgebra U q (u J ) is an augmented algebra but not a Hopf algebra. Moreover, the cohomology for U q (u J ) is We consider finite dimensional modules L q (λ) for U q (g), and calculate the cohomologyas a U q (l J )-module. In the case when q is a generic parameter our results recover an earlier result of Malikov [Mal] when g = sl(n) and u J = u is the unipotent radical of a Borel subalgebra. We also consider the case when q is specialized to an lth root of unity. In this situation, when l ≥ h − 1 (h being the Coxeter number of the underlying root system), we prove a version of Kostant's theorem for weights in the lowest alcove. These results can be regarded as an analog for quantum groups of results by Friedlander and Parshall [FP] and by Polo and Tilouine [PT, Corollary 2.10] for the cohomology of Lie algebras of reductive algebraic groups in prime characteristic.1.2. The paper is organized as follows. The basic information we shall n...

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Quantum linear groups

Cited by 225 publications

References 7 publications

Quantum Analogues of Schubert Varieties in the Grassmannian

Quantum Analogues of Schubert Varieties in the Grassmannian

Generic and $q$-Rational Representation Theory

An analog of Kostant’s theorem for the cohomology of quantum groups

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