Double affine Hecke algebras of rank 1 and affine cubic surfaces
By Alexei Oblomkov. 2003
72 citations recorded by the source corpus.
We study the algebraic properties of the five-parameter family $H(t_1,t_2,t_3,t_4;q)$ of double affine Hecke algebras of type $C^\vee C_1$. 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 $PGL_2(\Bbb Z)$ by $\Bbb Z$ 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 $q=e^\hbar$ provides its deformation quantization. Using that $H^2(C,\Bbb C)=\Bbb C^5$, we find that the Hochschild cohomology $HH^2(H)$ (for $q=e^\hbar$) 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. We also prove that the five-parameter family $H(t_1,t_2,t_3,t_4;q)$ of algebras yields the universal deformation of $q$-Weyl algebra crossed with ${\Bbb Z}_2$ and the family of cubic surfaces $C=C_{\underline{t}}$, $\underline{t}\in \CC^4_{\underline{t}}$ gives the universal deformation of the Poisson algebra $\CC[X^{\pm 1},P^{\pm 1}]^{{\Bbb ZZ}_2}$.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/a3a9c4a7617c28bc88ad202b9a7ab80d72ec8481
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Citations in the corpus
- Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra ? (2007, methodology): “The definition of spherical subalgebra for the DAHA of type (C∨1 , C1) was given by Oblomkov [10].”
- The finiteness conjecture for skein modules (2019, background): “Taking Hochschild homology, and recalling that HH0(Dq(H) Z2) ∼= Q(A)5 [73,75], we recover the computation, Sk(T 3) ∼= HH0(ZSL2(T 2)) ∼= Q(A)5 ⊕ Q(A) ⊕ Q(A) ⊕ Q(A) ⊕ Q(A), of [27,47] in a new way.”