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


SIGMA 7 (2011), 069, 24 pages      arXiv:1104.2813      http://dx.doi.org/10.3842/SIGMA.2011.069
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

The Universal Askey-Wilson Algebra

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

Received April 17, 2011, in final form July 09, 2011; Published online July 15, 2011

Abstract    (this is shortened html-version of the paper's abstract)
In 1992 A. Zhedanov introduced the Askey-Wilson algebra AW=AW(3) and used it to describe the Askey-Wilson polynomials. In this paper we introduce a central extension Δ of AW, obtained from AW by reinterpreting certain parameters as central elements in the algebra. We call Δ the universal Askey-Wilson algebra. We give a faithful action of the modular group PSL2(Z) on Δ as a group of automorphisms. We give a linear basis for Δ. We describe the center of Δ and the 2-sided ideal Δ[Δ,Δ]Δ. We discuss how Δ is related to the q-Onsager algebra.

Key words: Askey-Wilson relations; Leonard pair; modular group; q-Onsager algebra.

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References

  1. Alperin R.C., Notes: PSL2(Z)=Z2*Z3, Amer. Math. Monthly 100 (1993), 385-386.
  2. Aneva B., Tridiagonal symmetries of models of nonequilibrium physics, SIGMA 4 (2008), 056, 16 pages, arXiv:0807.4391.
  3. Askey R., Wilson J., Some basic hypergeometric polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319.
  4. Atakishiyev N.M., Klimyk A.U., Representations of the quantum algebra suq(1,1) and duality of q-orthogonal polynomials, in Algebraic Structures and their Representations, Contemp. Math., Vol. 376, Amer. Math. Soc., Providence, RI, 2005, 195-206.
  5. Atakishiyev M.N., Groza V., The quantum algebra Uq(su2) and q-Krawtchouk families of polynomials, J. Phys. A: Math. Gen. 37 (2004), 2625-2635.
  6. Atakishiyev M.N., Atakishiyev N.M., Klimyk A.U., Big q-Laguerre and q-Meixner polynomials and representations of the quantum algebra Uq(su1,1), J. Phys. A: Math. Gen. 36 (2003), 10335-10347, math.QA/0306201.
  7. Baseilhac P., An integrable structure related with tridiagonal algebras, Nuclear Phys. B 705 (2005), 605-619, math-ph/0408025.
  8. Baseilhac P., Deformed Dolan-Grady relations in quantum integrable models, Nuclear Phys. B 709 (2005), 491-521, hep-th/0404149.
  9. Baseilhac P., Koizumi K., A new (in)finite-dimensional algebra for quantum integrable models, Nuclear Phys. B 720 (2005), 325-347, math-ph/0503036.
  10. Baseilhac P., Koizumi K., A deformed analogue of Onsager's symmetry in the XXZ open spin chain, J. Stat. Mech. Theory Exp. 2005 (2005), no. 10, P10005, 15 pages, hep-th/0507053.
  11. Baseilhac P., The q-deformed analogue of the Onsager algebra: beyond the Bethe ansatz approach, Nuclear Phys. B 754 (2006), 309-328, math-ph/0604036.
  12. Baseilhac P., A family of tridiagonal pairs and related symmetric functions, J. Phys. A: Math. Gen. 39 (2006), 11773-11791, math-ph/0604035.
  13. Baseilhac P., Koizumi K., Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory, J. Stat. Mech. Theory Exp. 2007 (2007), no. 9, P09006, 27 pages, hep-th/0703106.
  14. Baseilhac P., Belliard S., Generalized q-Onsager algebras and boundary affine Toda field theories, Lett. Math. Phys. 93 (2010), 213-228, arXiv:0906.1215.
  15. Baseilhac P., Shigechi K., A new current algebra and the reflection equation, Lett. Math. Phys. 92 (2010), 47-65, arXiv:0906.1482.
  16. Bergman G., The diamond lemma for ring theory, Adv. Math. 29 (1978), 178-218.
  17. Carter R., Lie algebras of finite and affine type, Cambridge Studies in Advanced Mathematics, Vol. 96, Cambridge University Press, Cambridge, 2005.
  18. Ciccoli N., Gavarini F., A quantum duality principle for coisotropic subgroups and Poisson quotients, Adv. Math. 199 (2006), 104-135, math.QA/0412465.
  19. Curtin B., Spin Leonard pairs, Ramanujan J. 13 (2007), 319-332.
  20. Curtin B., Modular Leonard triples, Linear Algebra Appl. 424 (2007), 510-539.
  21. Etingof P., Ginzburg V., Noncommutative del Pezzo surfaces and Calabi-Yau algebras, J. Eur. Math. Soc. 12 (2010), 1371-1416, arXiv:0709.3593.
  22. Fairlie D.B., Quantum deformations of SU(2), J. Phys. A: Math. Gen. 23 (1990), L183-L187.
  23. Floratos E.G., Nicolis S., An SU(2) analogue of the Azbel-Hofstadter Hamiltonian, J. Phys. A: Math. Gen. 31 (1998), 3961-3975, hep-th/9508111.
  24. Granovskii Ya.A., Zhedanov A.S., Nature of the symmetry group of the 6j-symbol, Zh. Èksper. Teoret. Fiz. 94 (1988), 49-54 (English transl.: Soviet Phys. JETP 67 (1988), 1982-1985).
  25. Granovskii Ya.I., Lutzenko I.M., Zhedanov A.S., Mutual integrability, quadratic algebras, and dynamical symmetry, Ann. Physics 217 (1992), 1-20.
  26. Granovskii Ya.I., Zhedanov A.S., Linear covariance algebra for slq(2), J. Phys. A: Math. Gen. 26 (1993), L357-L359.
  27. Grünbaum F.A., Haine L., On a q-analogue of the string equation and a generalization of the classical orthogonal polynomials, in Algebraic Methods and q-Special Functions (Montréal, QC, 1996), CRM Proc. Lecture Notes, Vol. 22, Amer. Math. Soc., Providence, RI, 1999, 171-181.
  28. Havlícek M., Klimyk A.U., Posta S., Representations of the cyclically symmetric q-deformed algebra soq (3), J. Math. Phys. 40 (1999), 2135-2161, math.QA/9805048.
  29. Havlícek M., Posta S., On the classification of irreducible finite-dimensional representations of Uq'(so3) algebra, J. Math. Phys. 42 (2001), 472-500.
  30. 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.
  31. Ito T., Tanabe K., Terwilliger P., Some algebra related to P- and Q-polynomial association schemes, in Codes and Association Schemes (Piscataway, NJ, 1999), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Vol. 56, Amer. Math. Soc., Providence, RI, 2001, 167-192, math.CO/0406556.
  32. Ito T., Terwilliger P., Double affine Hecke algebras of rank 1 and the Z3-symmetric Askey-Wilson relations, SIGMA 6 (2010), 065, 9 pages, arXiv:1001.2764.
  33. Ito T., Terwilliger P., Tridiagonal pairs of q-Racah type, J. Algebra 322 (2009), 68-93, arXiv:0807.0271.
  34. Ito T., Terwilliger P., The augmented tridiagonal algebra, Kyushu J. Math. 64 (2010), 81-144, arXiv:0904.2889.
  35. Jordan D.A., Sasom N., Reversible skew Laurent polynomial rings and deformations of Poisson automorphisms, J. Algebra Appl. 8 (2009), 733-757, arXiv:0708.3923.
  36. Kalnins E., Miller W., Post S., Models for quadratic algebras associated with second order superintegrable systems in 2D, SIGMA 4 (2008), 008, 21 pages, arXiv:0801.2848.
  37. Koekoek R., Lesky P.A., Swarttouw R., Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  38. Koornwinder K.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, math.QA/0612730.
  39. Koornwinder K.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.
  40. Korovnichenko A., Zhedanov A., Classical Leonard triples, in Elliptic Integrable Systems (Kyoto, 2004), Editors M. Noumi and K. Takasaki, Rokko Lectures in Mathematics, no. 18, Kobe University, 2005, 71-84.
  41. Lavrenov A.N., Relativistic exactly solvable models, in Proceedings VIII International Conference on Symmetry Methods in Physics (Dubna, 1997), Phys. Atomic Nuclei 61 (1998), 1794-1796.
  42. Lavrenov A.N., On Askey-Wilson algebra, in Quantum Groups and Integrable Systems, II (Prague, 1997), Czechoslovak J. Phys. 47 (1997), 1213-1219.
  43. Lavrenov A.N., Deformation of the Askey-Wilson algebra with three generators, J. Phys. A: Math. Gen. 28 (1995), L503-L506.
  44. Nomura K., Terwilliger P., Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair, Linear Algebra Appl. 420 (2007), 198-207, math.RA/0605316.
  45. Oblomkov A., Double affine Hecke algebras of rank 1 and affine cubic surfaces, Int. Math. Res. Not. 2004 (2004), no. 18, 877-912, math.RT/0306393.
  46. Odake S., Sasaki R., Orthogonal polynomials from Hermitian matrices, J. Math. Phys. 49 (2008), 053503, 43 pages, arXiv:0712.4106.
  47. Odake S., Satoru R., Unified theory of exactly and quasiexactly solvable "discrete" quantum mechanics. I. Formalism, J. Math. Phys. 51 (2010), 083502, 24 pages, arXiv:0903.2604.
  48. Odesskii M., An analogue of the Sklyanin algebra, Funct. Anal. Appl. 20 (1986), 152-154.
  49. Rosenberg A.L., Noncommutative algebraic geometry and representations of quantized algebras, Mathematics and its Applications, Vol. 330, Kluwer Academic Publishers Group, Dordrecht, 1995.
  50. Rosengren H., Multivariable orthogonal polynomials as coupling coefficients for Lie and quantum algebra representations, Ph.D. Thesis, Centre for Mathematical Sciences, Lund University, Sweden, 1999.
  51. Rosengren H., An elementary approach to 6j-symbols (classical, quantum, rational, trigonometric, and elliptic), Ramanujan J. 13 (2007), 131-166, math.CA/0312310.
  52. Smith S.P., Bell A.D., Some 3-dimensional skew polynomial rings, Unpublished lecture notes, 1991.
  53. Terwilliger P., The subconstituent algebra of an association scheme. III, J. Algebraic Combin. 2 (1993), 177-210.
  54. Terwilliger P., Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear Algebra Appl. 330 (2001), 149-203, math.RA/0406555.
  55. Terwilliger P., Two relations that generalize the q-Serre relations and the Dolan-Grady relations, in Physics and Combinatorics 1999 (Nagoya), World Sci. Publ., River Edge, NJ, 2001, 377-398, math.QA/0307016.
  56. Terwilliger P., An algebraic approach to the Askey scheme of orthogonal polynomials, in Orthogonal Polynomials and Special Functions, Lecture Notes in Math., Vol. 1883, Springer, Berlin, 2006, 255-330, math.QA/0408390.
  57. Terwilliger P., Vidunas R., Leonard pairs and the Askey-Wilson relations, J. Algebra Appl. 3 (2004), 411-426, math.QA/0305356.
  58. Vidar M., Tridiagonal pairs of shape (1,2,1), Linear Algebra Appl. 429 (2008), 403-428, arXiv:0802.3165.
  59. Vidunas R., Normalized Leonard pairs and Askey-Wilson relations, Linear Algebra Appl. 422 (2007), 39-57, math.RA/0505041.
  60. Vidunas R., Askey-Wilson relations and Leonard pairs, Discrete Math. 308 (2008), 479-495, math.QA/0511509.
  61. Vinet L., Zhedanov A.S., Quasi-linear algebras and integrability (the Heisenberg picture), SIGMA 4 (2008), 015, 22 pages, arXiv:0802.0744.
  62. Vinet L., Zhedanov A.S., A "missing" family of classical orthogonal polynomials, J. Phys. A: Math. Theor. 44 (2011), 085201, 16 pages, arXiv:1011.1669.
  63. Vinet L., Zhedanov A.S., A limit q=−1 for the big q-Jacobi polynomials, Trans. Amer. Math. Soc., to appear, arXiv:1011.1429.
  64. Wiegmann P.B., Zabrodin A.V., Algebraization of difference eigenvalue equations related to Uq(sl2), Nuclear Phys. B 451 (1995), 699-724, cond-mat/9501129.
  65. Zhedanov A.S., "Hidden symmetry" of the Askey-Wilson polynomials, Theoret. and Math. Phys. 89 (1991), 1146-1157.
  66. Zhedanov A.S., Korovnichenko A., "Leonard pairs" in classical mechanics, J. Phys. A: Math. Gen. 35 (2002), 5767-5780.

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