Séminaire de Physique Théorique
The Universal Askey-Wilson algebraPaul Terwilliger (Univ. of Wisconsin-Madison)
Thursday 26 November 2015 14:00 - Tours - Salle 1180 (Bât E2)
Résumé :
The Askey-Wilson polynomials were introduced around 1985 and soon became a major topic in special functions. This topic became linked to representation theory around 1992 when A. Zhedanov introduced the Askey-Wilson algebra AW. The algebra AW is defined by generators and relations. The relations involve a scalar parameter $q$ and a handful of extra scalar parameters. We discuss a central extension of AW, denoted $\Delta_q$ and called the universal Askey-Wilson algebra. Roughly speaking, up to normalization $\Delta_q$ is obtained from AW by interpreting the extra parameters as central elements in the algebra. By construction $\Delta_q$ involves no parameters besides $q$. In this talk we relate $\Delta_q$ to the following objects: (i) Leonard pairs and Leonard triples of QRacah type; (ii) Q-polynomial distance-regular graphs; (iii) The modular group ${\rm PSL}_2(Z)$; (iv) The equitable presentation for the quantum group $U_q(sl_2)$; (v) The double affine Hecke algebra of type $(C_1^\vee, C_1)$. The talk will be elementary; we do not assume exposure to the above topics.
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