Harmonic analysis on the SU(2) dynamical quantum group

Abstract

Dynamical quantum groups were recently introduced by Etingof and Varchenko as an algebraic framework for studying the dynamical Yang-Baxter equation, which is precisely the Yang-Baxter equation satisfied by 6j-symbols. We investigate one of the simplest examples, generalizing the standard SU(2) quantum group. The matrix elements for its corepresentations are identified with Askey-Wilson polynomials, and the Haar measure with the Askey-Wilson measure. The discrete orthogonality of the matrix elements yield the orthogonality of q-Racah polynomials (or quantum 6j-symbols). The Clebsch-Gordan coefficients for representations and corepresentations are also identified with q-Racah polynomials. This results in new algebraic proofs of the Biedenharn-Elliott identity satisfied by quantum 6j-symbols.