The algebra of dual −1 Hahn polynomials and the Clebsch-Gordan problem of sl−1(2)

The algebra [12pt]{minimal}{document}H{document}H of the dual −1 Hahn polynomials is derived and shown to arise in the Clebsch-Gordan problem of sl−1(2). The dual −1 Hahn polynomials are the bispectral polynomials of a discrete argument obtained from the q → −1 limit of the dual q-Hahn polynomials. The Hopf algebra sl−1(2) has four generators including an involution, it is also a q → −1 limit of the quantum algebra slq(2) and furthermore, the dynamical algebra of the parabose oscillator. The algebra [12pt]{minimal}{document}H{document}H, a two-parameter generalization of [12pt]{minimal}{document}u(2){document}u(2) with an involution as additional generator, is first derived from the recurrence relation of the −1 Hahn polynomials. It is then shown that [12pt]{minimal}{document}H{document}H can be realized in terms of the generators of two added sl−1(2) algebras, so that the Clebsch-Gordan coefficients of sl−1(2) are dual −1 Hahn polynomials. An irreducible representation of [12pt]{minimal}{document}H{document}H involving five-diagonal matrices and connected to the difference equation of the dual −1 Hahn polynomials is constructed.