2 7 Ju n 20 03 DOUBLE AFFINE HECKE ALGEBRAS OF RANK 1 AND AFFINE CUBIC SURFACES
نویسنده
چکیده
We study the algebraic properties of the five-parameter family H(t1, t2, t3, t4; q) of double affine Hecke algebras of type C ∨ C1. This family generalizes Cherednik's double affine Hecke algebras of rank 1. It was introduced by Sahi and studied by Noumi and Stokman as an algebraic structure which controls Askey-Wilson polynomials. We show that if q = 1, then the spectrum of the center of H is an affine cubic surface C, obtained from a projective one by removing a triangle consisting of smooth points. Moreover, any such surface is obtained as the spectrum of the center of H for some values of parameters. This result allows one to give a simple geometric description of the action of an extension of P GL2(Z) by Z on the center of H. When C is smooth, it admits a unique algebraic symplectic structure, and the spherical subalgebra eHe of the algebra H for q = e provides its deformation quantization. Using that H 2 (C, C) = C 5 , we find that the Hochschild cohomology HH 2 (H) (for q = e) is 5-dimensional for generic parameter values. From this we deduce that the only deformations of H come from variations of parameters. This explains from the point of view of noncommutative geometry why one cannot add more parameters into the theory of Askey-Wilson polynomials.
منابع مشابه
ar X iv : m at h / 03 06 39 3 v 4 [ m at h . R T ] 2 4 N ov 2 00 3 DOUBLE AFFINE HECKE ALGEBRAS OF RANK 1 AND AFFINE CUBIC SURFACES
We study the algebraic properties of the five-parameter family H(t1, t2, t3, t4; q) of double affine Hecke algebras of type C ∨ C1. This family generalizes Cherednik's double affine Hecke algebras of rank 1. It was introduced by Sahi and studied by Noumi and Stokman as an algebraic structure which controls Askey-Wilson polynomials. We show that if q = 1, then the spectrum of the center of H is ...
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0 O ct 2 00 3 DOUBLE AFFINE HECKE ALGEBRAS OF RANK 1 AND AFFINE CUBIC SURFACES
We study the algebraic properties of the five-parameter family H(t1, t2, t3, t4; q) of double affine Hecke algebras of type C ∨ C1. This family generalizes Cherednik's double affine Hecke algebras of rank 1. It was introduced by Sahi and studied by Noumi and Stokman as an algebraic structure which controls Askey-Wilson polynomials. We show that if q = 1, then the spectrum of the center of H is ...
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We study the algebraic properties of the five-parameter family H(t1, t2, t3, t4; q) of double affine Hecke algebras of type C ∨ C1. This family generalizes Cherednik's double affine Hecke algebras of rank 1. It was introduced by Sahi and studied by Noumi and Stokman as an algebraic structure which controls Askey-Wilson polynomials. We show that if q = 1, then the spectrum of the center of H is ...
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The goal of this paper is to define a new class of objects which we call triple affine Artin groups and to relate them with Cherednik’s double affine Hecke algebras. This has as immediate consequences new and simple descriptions of double affine Weyl and Artin groups, the double affine Hecke algebras as well as the corresponding elliptic objects. We also recover in an transparent and elementary...
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تاریخ انتشار 2004