Sklyanin algebras and Hilbert schemes of points

Nevins, Thomas; Stafford, J. T.

doi:10.1016/j.aim.2006.06.009

Cited by 36 publications

(95 citation statements)

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“…This result was recently proved for almost all A by Nevins and Stafford [27], using deformation theoretic methods and the known commutative case. In the case where A is the homogenization of the first Weyl algebra this result was also proved by Wilson in [37].…”

Section: Remark 14mentioning

confidence: 57%

“…The Hilbert scheme Hilb n (P 2 q ) was constructed in [27] (see also [17] for a somewhat less general result and [10] in the case where A is the homogenization of the first Weyl algebra). The definition of Hilb n (P 2 q ) is not entirely straightforward since in general P 2 q will have very few zero dimensional non-commutative subschemes (see [31]), so a different approach is needed.…”

Section: Below)mentioning

confidence: 99%

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Ideal classes of three dimensional Artin–Schelter regular algebras

Naeghel

Bergh

2005

Journal of Algebra

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We determine the possible Hilbert functions of graded rank one torsion free modules over three dimensional Artin-Schelter regular algebras. It turns out that, as in the commutative case, they are related to Castelnuovo functions. From this we obtain an intrinsic proof that the space of torsion free rank one modules on a non-commutative P 2 is connected. A different proof of this fact, based on deformation theoretic methods and the known commutative case has recently been given by Nevins and Stafford [Sklyanin algebras and Hilbert schemes of points, math.AG/0310045]. For the Weyl algebra it was proved by Wilson [Invent. Math. 133 (1) (1998) 1-41].

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Section: Remark 14mentioning

confidence: 57%

Section: Below)mentioning

confidence: 99%

Ideal classes of three dimensional Artin–Schelter regular algebras

Naeghel

Bergh

2005

Journal of Algebra

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“…[n] as well as with their non-commutative counterparts proposed in [43]. We can argue that the proposed in [9] quantization scheme for the fraction fields can be applied to the NC surface (P S \E) and to "extend" the deformation to the NC Hilbert scheme (P S \E)…”

Section: Discussion and Future Problemsmentioning

confidence: 99%

“…[n] introducing in [43]. The proposition 3.1 from [12] then should in principle to give an NC integrable Beauville-Mukai system on (P S \E)…”

Section: Discussion and Future Problemsmentioning

confidence: 99%

Integrable systems associated with elliptic algebras

Odesskii

Rubtsov

2008

IRMA Lectures in Mathematics and Theoretical Physics

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Abstract. We construct some new Integrable Systems (IS) both classical and quantum associated with elliptic algebras. Our constructions are partly based on the algebraic integrability mechanism given by the existence of commuting families in skew fields and partly -on the internal properties of the elliptic algebras and their representations. We give some examples to make an evidence how these IS are related to previously studied.Introduction. This paper is an attempt to establish a direct connection between two close subjects of modern Mathematical Physics -the theory of Integrable Systems (IS) and the Elliptic Algebras. The aim of this connection is two-fold: we clarify some our recent results in the both domains and fill the natural gap proving that some large class of the Elliptic Algebras carries in fact the structure of an IS.We will start with a short account in the subject of the story and will describe briefly a type of the IS's under cosideration.In [12], B.Enriquez and second author had proposed a construction of commuting families of elements in skew fields. They explained how to use this construction in Poisson fraction fields to give an another proof of the integrability of Beauville -Mukai integrable systems associated with a K3 surface S ([1]).The Beauville-Mukai systems had appeared as the Lagrangian fibrations which have the formis the Hilbert scheme of g points of S, equipped with a symplectic structure introduced in [19], and L is a line bundle on S. Later, the authors of [8] explained that these systems are natural deformations of the "separated" (in the sense of [13]) versions of Hitchin's integrable systems, more precisely, of their description in terms of spectral curves (already present in [15]). Beauville-Mukai systems can be generalized to surfaces with a Poisson structure (see [3]). When S is the canonical cone Cone(C) of an algebraic curve C these systems coincide with the separated version of Hitchin's systems. The paper [12] shows how the commuting families construction provides a quantization of these systems on the canonical cone.A quantization of Hitchin's system was proposed in [2]. It seems interesting to construct quantization of Beauville-Mukai systems and to compare it with Beilinson -Drinfeld quantization. We had conjectured the correspondence between the results of [2] and a quantization of fraction fields in [9]. A part of this program

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“…The authors became interested in the incidence problem while they were studying the deformations of the Hilbert schemes of P 2 which come from non-commutative geometry, see [16,6,7].…”

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confidence: 99%

On incidence between strata of the Hilbert scheme of points on $$\mathbb{P}^{2}$$

Naeghel

Bergh

2006

Math. Z.

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The Hilbert scheme of n points in the projective plane has a natural stratification obtained from the associated Hilbert series. In general, the precise inclusion relation between the closures of the strata is still unknown. Guerimand, Ph.D Thesis, Universite'de Nice, 2002 studied this problem for strata whose Hilbert series are as close as possible. Preimposing a certain technical condition he obtained necessary and sufficient conditions for the incidence of such strata. In this paper we present a new approach, based on deformation theory, to Guerimand's result. This allows us to show that the technical condition is not necessary.

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Sklyanin algebras and Hilbert schemes of points

Cited by 36 publications

References 57 publications

Ideal classes of three dimensional Artin–Schelter regular algebras

Ideal classes of three dimensional Artin–Schelter regular algebras

Integrable systems associated with elliptic algebras

On incidence between strata of the Hilbert scheme of points on $$\mathbb{P}^{2}$$

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