The dynamical Yang-Baxter equation and Wigner 9j-symbols

Abstract

The quantum dynamical Yang-Baxter equation (QDYBE) plays a central role in some recent work on quantum groups. Babelon, Bernard & Billey found a way to construct solutions of the QDYBE using certain ``twisted boundaries'' in a quantum group. In this talk I will describe how these objects arise naturally in harmonic analysis on the SL(2) quantum group (using Koornwinder's method of twisted primitive elements). In fact, they play the role of ``points'' in the quantum group. This leads to an algebraic approach to the QDYBE which is quite different from the ones discussed in the lectures by Koornwinder and Koelink.

On the level of special functions, this yields an interpretation of $q$-Racah polynomials (quantum 6j-symbols) which is more elementary than the standard one, ivolving a one-fold rather than a three-fold tensor product. For two-fold tensor products one similarly obtains general 9j-symbols (which are two-variable analogues of q-Racah polynomials) as natural q-analogues of elementary products of two Krawtchouk polynomials. In fact, four seemingly different kinds of two-variable $q$-Racah polynomials appear.