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


SIGMA 7 (2011), 099, 26 pages      arXiv:1107.3544      http://dx.doi.org/10.3842/SIGMA.2011.099

The Universal Askey-Wilson Algebra and the Equitable Presentation of Uq(sl2)

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

Received July 19, 2011, in final form October 10, 2011; Published online October 25, 2011; Misprint in Lemma 7.1 is corrected March 16, 2012

Abstract    (this is shortened html-version of the paper's abstract)
Around 1992 A. Zhedanov introduced the Askey-Wilson algebra AW(3). Recently we introduced a central extension Δ of AW(3) called the universal Askey-Wilson algebra. In this paper we discuss a connection between Δ and the quantum algebra Uq(sl2). Our main result is an algebra injection from Δ into a relative of Uq(sl2); the relative is obtained from Uq(sl2) by adjoining three mutually commuting indeterminates. We describe the injection using the equitable presentation of Uq(sl2).

Key words: Askey-Wilson relations; Leonard pair; Casimir element.

pdf (534 kb)   tex (27 kb)       [previous version:  pdf (534 kb)   tex (27 kb)]

References

  1. Alnajjar H., Leonard pairs from the equitable generators of Uq(sl2), Dirasat Pure Sciences, Vol. 37, University of Jordan, 2010, available at http://www.ju.edu.jo/sites/Academic/h.najjar.
  2. Alnajjar H., Leonard pairs associated with the equitable generators of the quantum algebra Uq(sl2), Linear Multilinear Algebra 59 (2011), 1127-1142.
  3. Alperin R.C., Notes: PSL2(Z)=Z2*Z3, Amer. Math. Monthly 100 (1993), 385-386.
  4. Askey R., Wilson J., Some basic hypergeometric polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319.
  5. Baseilhac P., An integrable structure related with tridiagonal algebras, Nuclear Phys. B 705 (2005), 605-619, math-ph/0408025.
  6. Granovski Ya.I., Zhedanov A.S., Linear covariance algebra for slq(2), J. Phys. A: Math. Gen. 26 (1993), L357-L359.
  7. Ismail M., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2009.
  8. 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.
  9. Ito T., Terwilliger P., Weng C.-W., The quantum algebra Uq(sl2) and its equitable presentation, J. Algebra 298 (2006), 284-301, math.QA/0507477.
  10. Jantzen J.C., Lectures on quantum groups, Graduate Studies in Mathematics, Vol. 6, American Mathematical Society, Providence, RI, 1996.
  11. Koekoek R., Lesky P.A., Swarttouw R., Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  12. 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, math.QA/0612730.
  13. 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.
  14. Odake S., Sasaki R., Unified theory of exactly and quasiexactly solvable "discrete" quantum mechanics. I. Formalism, J. Math. Phys. 51 (2010), 083502, 24 pages, arXiv:0903.2604.
  15. 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.
  16. Terwilliger P., Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array, Des. Codes Cryptogr. 34 (2005), 307-332, math.RA/0306291.
  17. Terwilliger P., The universal Askey-Wilson algebra, SIGMA 7 (2011), 069, 24 pages, arXiv:1104.2813.
  18. Terwilliger P., Vidunas R., Leonard pairs and the Askey-Wilson relations, J. Algebra Appl. 3 (2004), 411-426, math.QA/0305356.
  19. 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.
  20. Zhedanov A.S., "Hidden symmetry" of the Askey-Wilson polynomials, Theoret. and Math. Phys. 89 (1991), 1146-1157.

Previous article   Next article   Contents of Volume 7 (2011)