SYGN 2012 Abstracts - Paul Terwilliger

The universal Askey-Wilson algebra
Paul Terwilliger (University of Wisconsin-Madison, USA)

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 introduce 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 $PSL_2(Z)$;
(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 very elementary; we do not assume exposure to the above topics.