### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 047, 40 pages      arXiv:1202.4673      https://doi.org/10.3842/SIGMA.2013.047

### The Universal Askey-Wilson Algebra and DAHA of Type $(C_1^{\vee},C_1)$

Paul Terwilliger
Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA

Received December 22, 2012, in final form July 07, 2013; Published online July 15, 2013

Abstract
Let $\mathbb F$ denote a field, and fix a nonzero $q\in\mathbb F$ such that $q^4\not=1$. The universal Askey-Wilson algebra $\Delta_q$ is the associative $\mathbb F$-algebra defined by generators and relations in the following way. The generators are $A$, $B$, $C$. The relations assert that each of $A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}}$, $B+\frac{qCA-q^{-1}AC}{q^2-q^{-2}}$, $C+\frac{qAB-q^{-1}BA}{q^2-q^{-2}}$ is central in $\Delta_q$. The universal DAHA $\hat H_q$ of type $(C_1^\vee,C_1)$ is the associative $\mathbb F$-algebra defined by generators $\lbrace t^{\pm1}_i\rbrace_{i=0}^3$ and relations (i) $t_i t^{-1}_i=t^{-1}_i t_i=1$; (ii) $t_i+t^{-1}_i$ is central; (iii) $t_0t_1t_2t_3=q^{-1}$. We display an injection of $\mathbb F$-algebras $\psi:\Delta_q\to\hat H_q$ that sends $A\mapsto t_1t_0+(t_1t_0)^{-1}$, $B\mapsto t_3t_0+(t_3t_0)^{-1}$, $C\mapsto t_2t_0+(t_2t_0)^{-1}$. For the map $\psi$ we compute the image of the three central elements mentioned above. The algebra $\Delta_q$ has another central element of interest, called the Casimir element $\Omega$. We compute the image of $\Omega$ under $\psi$. We describe how the Artin braid group $B_3$ acts on $\Delta_q$ and $\hat H_q$ as a group of automorphisms. We show that $\psi$ commutes with these $B_3$ actions. Some related results are obtained.

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References

1. Alperin R.C., ${\rm PSL}_2(Z) = Z_2 \star Z_3$, Amer. Math. Monthly 100 (1993), 385-386.
2. Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55 pages.
3. Bergman G.M., The diamond lemma for ring theory, Adv. Math. 29 (1978), 178-218.
4. Cherednik I., Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald's operators, Int. Math. Res. Not. (1992), 171-180.
5. Ion B., Sahi S., Triple groups and Cherednik algebras, in Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, Amer. Math. Soc., Providence, RI, 2006, 183-206, math.QA/0304186.
6. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2009.
7. Ito T., Terwilliger P., Double affine Hecke algebras of rank 1 and the ${\mathbb Z}_3$-symmetric Askey-Wilson relations, SIGMA 6 (2010), 065, 9 pages, arXiv:1001.2764.
8. Ito T., Terwilliger P., Weng C.-W., The quantum algebra $U_q(\mathfrak{sl}_2)$ and its equitable presentation, J. Algebra 298 (2006), 284-301, math.QA/0507477.
9. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
10. Koornwinder T.H., The relationship between Zhedanov's algebra AW(3) and the double affine Hecke algebra in the rank one case, SIGMA 3 (2007), 063, 15 pages, arXiv:math.QA/0612730.
11. Koornwinder T.H., Zhedanov's algebra AW(3) and the double affine Hecke algebra in the rank one case. II. The spherical subalgebra, SIGMA 4 (2008), 052, 17 pages, arXiv:0711.2320.
12. Korovnichenko A., Zhedanov A., Classical Leonard triples, in Elliptic Integrable Systems (2004, Kyoto), Editors M. Noumi, K. Takasaki, Rokko Lectures in Mathematics, no. 18, Kobe University, 2005, 71-84.
13. Oblomkov A., Double affine Hecke algebras of rank 1 and affine cubic surfaces, Int. Math. Res. Not. 2004 (2004), 877-912, math.RT/0306393.
14. Sahi S., Nonsymmetric Koornwinder polynomials and duality, Ann. of Math. (2) 150 (1999), 267-282, q-alg/9710032.
15. Terwilliger P., The universal Askey-Wilson algebra, SIGMA 7 (2011), 069, 24 pages, arXiv:1104.2813.
16. Terwilliger P., The universal Askey-Wilson algebra and the equitable presentation of $U_q(\mathfrak{sl}_2)$, SIGMA 7 (2011), 099, 26 pages, arXiv:1107.3544.
17. Terwilliger P., Vidunas R., Leonard pairs and the Askey-Wilson relations, J. Algebra Appl. 3 (2004), 411-426, math.QA/0305356.
18. Wiegmann P.B., Zabrodin A.V., Algebraization of difference eigenvalue equations related to $U_q({\rm sl}_2)$, Nuclear Phys. B 451 (1995), 699-724, cond-mat/9501129.
19. Zhedanov A.S., "Hidden symmetry" of Askey-Wilson polynomials, Theoret. and Math. Phys. 89 (1991), 1146-1157.