Noncommutative del Pezzo surfaces and Calabi-Yau algebras

Etingof, Pavel; Ginzburg, Victor

doi:10.4171/jems/235

Cited by 40 publications

(60 citation statements)

References 34 publications

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“…By Proposition 3, the commutation relations (3.23), (3.27) and (3.28) -or equivalently (3.31)provide a presentation of the AW algebra with basic generators X, Y, W that differs from the original one [19]. Let us point out that the form of the defining relations in this presentation looks very close to the structure of the relations occuring in the three dimensional Calabi-Yau and Sklyanin algebras [3], [5]. In this presentation, let us observe that the r.h.s.…”

Section: )mentioning

confidence: 76%

The Heun–Askey–Wilson Algebra and the Heun Operator of Askey–Wilson Type

Baseilhac

Tsujimoto

Vinet

et al. 2019

Ann. Henri Poincaré

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The Heun-Askey-Wilson algebra is introduced through generators {X, W} and relations. These relations can be understood as an extension of the usual Askey-Wilson ones. A central element is given, and a canonical form of the Heun-Askey-Wilson algebra is presented. A homomorphism from the Heun-Askey-Wilson algebra to the Askey-Wilson one is identified. On the vector space of the polynomials in the variable x = z + z −1 , the Heun operator of Askey-Wilson type realizing W can be characterized as the most general second order q-difference operator in the variable z that maps polynomials of degree n in x = z + z −1 into polynomials of degree n + 1.1 2 PASCAL BASEILHAC, SATOSHI TSUJIMOTO, LUC VINET, AND ALEXEI ZHEDANOV operator and shown to be identical to the AHO connected to the big q-Jacobi polynomials. A special case readily seen to be the AHO of the little q-Jacobi polynomials has been called the little q-Heun operator. The big and little q-Heun operators were recognized to be two of the q-Heun operators introduced by Takemura [15] as degenerations of the Ruijenaars-van Diejen operators [14,4] thereby adding to the understanding of these topics. We now follow up with possibly the most important case: the q-analog of the Heun equation on the Askey grid.The tridiagonalization method has been used to inform the theory of higher polynomials (in the Askey scheme) from the lower ones [10,11,6]. The basic idea is to built the defining operator of the higher polynomials as special bilinear combinations of the bispectral operators of the lower polynomials. This splits the parameters of the higher polynomials in two classes: on the one hand, the set of parameters pertaining to the lower polynomials and on the other, the parameters entering the tridiagonalization. From an algebraic perspective, such special tridiagonalizations effect embeddings of the quadratic algebra associated to the higher polynomials into the algebra of the lower system. This is done by replacing the defining operator of the lower polynomials by the tridiagonalized one. In the q-sector, one has the Askey-Wilson (AW) algebra [19] at the highest level. Obviously the general tridiagonalization that yields AHOs takes one outside the framework of orthogonal polynomials. It is of interest to provide an algebraic picture of this situation and to unravel the structure that extends the AW algebra to an HAW algebra. This will also be done here.The outline of this paper is as follows. In Section 2, we introduce the Heun-Askey-Wilson (HAW) algebra. This algebra generalizes the Askey-Wilson algebra, whose defining relations can be recovered for certain choices of the structure constants. A central element is constructed. Also, for generic values of the deformation parameter q, a canonical presentation of the HAW algebra is exhibited. In Section 3, we study the relation between the HAW and the Askey-Wilson algebras. An explicit homomorphism from the HAW algebra to the Askey-Wilson one is given, inducing a new presentation that is closely related with the "Z 3 symmetri...

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Section: )mentioning

confidence: 76%

The Heun–Askey–Wilson Algebra and the Heun Operator of Askey–Wilson Type

Baseilhac

Tsujimoto

Vinet

et al. 2019

Ann. Henri Poincaré

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“…inherits the Poisson algebra structure [6]. In the case of the cubic Q defined by (1.5), the natural Poisson brackets in the Painlevé six monodromy manifold are given by…”

Section: Classification Of Quadratic Transformations On the Monodromymentioning

confidence: 99%

Cubic and Quartic Transformations of the Sixth Painlevé Equation in Terms of Riemann–Hilbert Correspondence

Mazzocco

Vidūnas

2012

Stud Appl Math

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Abstract. A starting point of this paper is a classification of quadratic polynomial transformations of the monodromy manifold for the 2 × 2 isomonodromic Fuchsian systems associated to the Painlevé VI equation. Up to birational automorphisms of the monodromy manifold, we find three transformations. Two of them are identified as the action of known quadratic or quartic transformations of the Painlevé VI equation. The third transformation of the monodromy manifold gives a new transformation of degree 3 of Picard's solutions of Painlevé VI.

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“…This condition forces the quasi-homogeneous polynomial h to be a nonzero constant. Since its degree is equal to k − a − b − c, we must have k = a + b + c. The first statement now follows from K. Saito's classification of such quasi-homogeneous polynomials [43]; see also the formulae in [19,Proposition 2.3.2].…”

Section: Whenmentioning

confidence: 93%

Elliptic singularities on log symplectic manifolds and Feigin–Odesskii Poisson brackets

Pym¹

2017

Compositio Math.

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A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface. The possible singularities of such a hypersurface are heavily constrained. We introduce the notion of an elliptic point of a log symplectic structure, which is a singular point at which a natural transversality condition involving the modular vector field is satisfied, and we prove a local normal form for such points that involves the simple elliptic surface singularities E6, E7 and E8. Our main application is to the classification of Poisson brackets on Fano fourfolds. For example, we show that Feigin and Odesskii's Poisson structures of type q5,1 are the only log symplectic structures on projective four-space whose singular points are all elliptic.

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Noncommutative del Pezzo surfaces and Calabi-Yau algebras

Cited by 40 publications

References 34 publications

The Heun–Askey–Wilson Algebra and the Heun Operator of Askey–Wilson Type

The Heun–Askey–Wilson Algebra and the Heun Operator of Askey–Wilson Type

Cubic and Quartic Transformations of the Sixth Painlevé Equation in Terms of Riemann–Hilbert Correspondence

Elliptic singularities on log symplectic manifolds and Feigin–Odesskii Poisson brackets

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