Quantum deformation of the flag variety

Fioresi, R.

doi:10.1080/00927879908826782

Cited by 16 publications

(25 citation statements)

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“…. Technically this is realized, along the line of [17,18,19,20,21] (see also [22,23,24] for a generalization to the super setting) and [25,26]. We also speculate on the interpretation of the "new" parameter ℓ , showing that the product of two ℓ commuting (or linked) quantum planes (or quantum spinors in the physical language [27]) reproduces the deformed algebra of the complex q-linked Minkowski space.…”

Section: Introductionmentioning

confidence: 84%

The q-linked complex Minkowski space, its real forms and deformed isometry groups

Fioresi

Latini

Marrani

2019

Int. J. Geom. Methods Mod. Phys.

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We establish duality between real forms of the quantum deformation of the 4-dimensional orthogonal group studied by Fioresi et al. in [15] and the classification work made by Borowiec et al. in [12]. Classically these real forms are the isometry groups of R 4 equipped with Euclidean, Kleinian or Lorentzian metric. A general deformation, named q-linked, of each of these spaces is then constructed, together with the coaction of the corresponding isometry group.

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Section: Introductionmentioning

confidence: 84%

The q-linked complex Minkowski space, its real forms and deformed isometry groups

Fioresi

Latini

Marrani

2019

Int. J. Geom. Methods Mod. Phys.

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“…Later in [4] is considered also the case of commutation relations among quantum minors with possibly different size, but with the first columns in common. In both cases the purpose of the formula is to obtain a quantum deformation of the grassmannian and flag manifolds.…”

Section: Introductionmentioning

confidence: 99%

Commutation relations among quantum minors in Oq(Mn(k))

Fioresi

2004

Journal of Algebra

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“…[1,8,13,16,17,23,26,29,35,38,39,40]. Around the same time, an approach to noncommutative projective algebraic geometry was initiated by Artin et al [4,5] and Artin and Van den Bergh [6], and considerably developed since (see e.g.…”

Section: Introductionmentioning

confidence: 99%

“Classical” Flag Varieties for Quantum Groups: The Standard Quantum SL(n,C)

Ohn

2002

Advances in Mathematics

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We suggest a possible programme to associate geometric ''flag-like'' data to an arbitrary simple quantum group, in the spirit of the noncommutative algebraic geometry developed by Artin, Tate, and Van den Bergh. We then carry out this programme for the standard quantum SLðnÞ of Drinfel'd and Jimbo, where the varieties involved are certain T-stable subvarieties of the (ordinary) flag variety. # 2002 Elsevier Science (USA) INTRODUCTIONThe study of quantum analogues of flag varieties, first suggested by Manin [31], has been undertaken during the past decade by several authors, from various points of view; see e.g. [1,8,13,16,17,23,26,29,35,38,39,40]. Around the same time, an approach to noncommutative projective algebraic geometry was initiated by Artin et al. [4,5] and Artin and Van den Bergh [6], and considerably developed since (see e.g. [3, 7, 9, 24, 28, 37, 41, 42, 44]). One attractive feature of their approach is the association of actual geometric data to certain classes of graded noncommutative algebras.The present work is an attempt to study quantum flag varieties from this point of view. As a consequence, our ''quantum flag varieties'' will be actual varieties (with some bells and whistles).Recall the original idea of [4][5][6]. If A is the homogeneous coordinate ring of a projective scheme E; then the points of E are in one-to-one correspondence with the isomorphism classes of the so-called point modules of A; i.e. N-graded cyclic A-modules P such that dim P n ¼ 1 for all n: Now if A is an N-graded noncommutative algebra, one may still try to parametrize its point modules by the points of some projective scheme E: Of course, one cannot hope to reconstruct A from E alone, but there is now an additional ingredient: the shift operation s : P/P½1; where P½1 is the N-graded A-module defined by P½1 n :¼ P nþ1 : (When A is commutative, this shift is trivial: P½1 ' P for every point module P:) Assume that s may be viewed as an automorphism of E: one may then hope, at least in ''good'' cases, to recover A from the triple ðE; s; LÞ; where L is the line bundle over E defined by its embedding into a projective space. The first step of this recovery is the construction of the twisted homogeneous coordinate ring BðE; s; LÞ of a triple ðE; s; LÞ; defined in [6] as follows:(where L s denotes the pullback of L along s), the multiplication being given by ab :¼ a b s m for all a 2 B m ; b 2 B n : (When s is the identity, this algebra coincides, in high degree, with the (commutative) homogeneous coordinate ring of E w.r.t. the polarization L:) If the triple ðE; s; LÞ comes from an algebra A as above, the second step then consists in analysing the canonical morphism A ! BðE; s; LÞ: The initial success of this method has been a complete study of all regular algebras of dimension three [4] (where the kernel of A ! BðE; s; LÞ turns out to be generated by a single element of degree three).The present paper is organized as follows.In Part I, we give a general outline of a possible theory of flag varieties for quantum groups, usi...

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Quantum deformation of the flag variety

Cited by 16 publications

References 10 publications

The q-linked complex Minkowski space, its real forms and deformed isometry groups

The q-linked complex Minkowski space, its real forms and deformed isometry groups

Commutation relations among quantum minors in Oq(Mn(k))

“Classical” Flag Varieties for Quantum Groups: The Standard Quantum SL(n,C)

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