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

2012/07/31 by Vincent X Genest, Luc Vinet, Alexei Zhedanov · 1 citation
Physics and Astronomy · Mathematics · #math-ph #math.MP #math.QA

paper · pdf · doi:10.1063/1.4790417

published as J. Math. Phys. 54, 02356 (2013) · 15 pages, Some minor changes from version #1

arxiv created 2013/02/12 · arxiv updated 2013/02/13

Abstract

The algebra 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 a 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 sl_q(2) and furthermore, the dynamical algebra of the parabose oscillator. The algebra H, a two-parameter generalization of 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 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 H involving five-diagonal matrices and connected to the difference equation of the dual -1 Hahn polynomials is constructed.

Cited by

• Verma modules and finite-dimensional irreducible modules of the universal Askey–Wilson algebra at roots of unity

Related

• Polynomials of the Askey scheme as Clebsch-Gordan coefficients
• Dual -1 Hahn polynomials: "classical" polynomials beyond the Leonard duality
• Hidden symmetry of Hahn problem for sl_q(2)
• Constructing bispectral dual Hahn polynomials
• A unified algebraic underpinning for the Hahn polynomials and rational functions