The Universal Askey–Wilson Algebra and the Equitable Presentation of $U_q(\mathfrak{sl}_2)$

Paul Terwilliger

Symmetry, integrability and geometry: methods and applications, Tome 7 (2011)

Résumé

Let $\mathbb F$ denote a field, and fix a nonzero $q\in\mathbb F$ such that $q^4\ne=1$. The universal Askey–Wilson algebra is the associative $\mathbb F$-algebra $\Delta=\Delta_q$ defined by generators and relations in the following way. The generators are $A$, $B$, $C$. The relations assert that each of $$ A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}}, \qquad B + \frac{qCA-q^{-1}AC}{q^2-q^{-2}}, \qquad C + \frac{qAB-q^{-1}BA}{q^2-q^{-2}} $$ is central in $\Delta$. In this paper we discuss a connection between $\Delta$ and the $\mathbb F$-algebra $U=U_q(\mathfrak{sl}_2)$. To summarize the connection, let $a$, $b$, $c$ denote mutually commuting indeterminates and let $\mathbb F \lbrack a^{\pm 1}, b^{\pm 1}, c^{\pm 1}\rbrack$ denote the $\mathbb F$-algebra of Laurent polynomials in $a$, $b$, $c$ that have all coefficients in $\mathbb F$. We display an injection of $\mathbb F$-algebras $ \Delta\to U \otimes_\mathbb F \mathbb F \lbrack a^{\pm 1}, b^{\pm 1}, c^{\pm 1}\rbrack$. For this injection we give the image of $A$, $B$, $C$ and the above three central elements, in terms of the equitable generators for $U$. The algebra $\Delta $ has another central element of interest, called the Casimir element $\Omega$. One significance of $\Omega$ is the following. It is known that the center of $\Delta$ is generated by $\Omega$ and the above three central elements, provided that $q$ is not a root of unity. For the above injection we give the image of $\Omega$ in terms of the equitable generators for $U$. We also use the injection to show that $\Delta$ contains no zero divisors.

Détail

Keywords: Askey–Wilson relations, Leonard pair
Mots-clés : Casimir element.

Bibliographie

[1] Alnajjar H., Leonard pairs from the equitable generators of $U_q(\mathfrak{sl}_2)$, Dirasat Pure Sciences, 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 $U_q(\mathfrak{sl}_2)$”, Linear Multilinear Algebra, 59 (2011), 1127–1142 | DOI

[3] Alperin R.C., “Notes: $\mathrm{PSL}_2(\mathbb Z)= \mathbb Z_2\star\mathbb Z_3$”, Amer. Math. Monthly, 100 (1993), 385–386 | DOI | MR | Zbl

[4] Askey R., Wilson J., “Some basic hypergeometric polynomials that generalize Jacobi polynomials”, Mem. Amer. Math. Soc., 54, no. 319 (1985) | MR

[5] Baseilhac P., “An integrable structure related with tridiagonal algebras”, Nuclear Phys. B, 705 (2005), 605–619 ; arXiv: math-ph/0408025 | DOI | MR | Zbl

[6] Granovskiĭ Ya.I., Zhedanov A.S., “Linear covariance algebra for $\mathrm{sl}_{q(2)}$”, J. Phys. A: Math. Gen., 26 (1993), L357–L359 | DOI | MR | Zbl

[7] Ismail M., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, 98, Cambridge University Press, Cambridge, 2009 | MR | Zbl

[8] 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 pp. ; arXiv: 1001.2764 | DOI | MR | Zbl

[9] 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 ; arXiv: math.QA/0507477 | DOI | MR | Zbl

[10] Jantzen J.C., Lectures on quantum groups, Graduate Studies in Mathematics, 6, American Mathematical Society, Providence, RI, 1996 | MR | Zbl

[11] Koekoek R., Lesky P.A., Swarttouw R., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010 | DOI | MR | Zbl

[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 pp. ; arXiv: math.QA/0612730 | DOI | MR | Zbl

[13] Lavrenov A.N., “Relativistic exactly solvable models”, Proceedings VIII International Conference on Symmetry Methods in Physics (Dubna, 1997), Phys. Atomic Nuclei, 61, 1998, 1794–1796 | MR

[14] Odake S., Sasaki R., “Unified theory of exactly and quasiexactly solvable “discrete” quantum mechanics. I. Formalism”, J. Math. Phys., 51 (2010), 083502, 24 pp. ; arXiv: 0903.2604 | DOI | MR

[15] Terwilliger P., “Two linear transformations each tridiagonal with respect to an eigenbasis of the other”, Linear Algebra Appl., 330 (2001), 149–203 ; arXiv: math.RA/0406555 | DOI | MR | Zbl

[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 ; arXiv: math.RA/0306291 | DOI | MR | Zbl

[17] Terwilliger P., “The universal Askey–Wilson algebra”, SIGMA, 7 (2011), 069, 24 pp. ; arXiv: 1104.2813 | DOI

[18] Terwilliger P., Vidunas R., “Leonard pairs and the Askey–Wilson relations”, J. Algebra Appl., 3 (2004), 411–426 ; arXiv: math.QA/0305356 | DOI | MR | Zbl

[19] Wiegmann P.B., Zabrodin A.V., “Algebraization of difference eigenvalue equations related to $U_q(sl_2)$”, Nuclear Phys. B, 451 (1995), 699–724 ; arXiv: cond-mat/9501129 | DOI | MR | Zbl

[20] Zhedanov A.S., ““Hidden symmetry” of the Askey–Wilson polynomials”, Theoret. and Math. Phys., 89 (1991), 1146–1157 | DOI | MR | Zbl