The q-Onsager Algebra and the Universal Askey-Wilson Algebra

Abstract

Recently Pascal Baseilhac and Stefan Kolb obtained a PBW basis for the q-Onsager algebra Oq. They defined the PBW basis elements recursively, and it is obscure how to express them in closed form. To mitigate the difficulty, we bring in the universal Askey-Wilson algebra Δq. There is a natural algebra homomorphism ♮: Oq → Δq. We apply ♮ to the above PBW basis, and express the images in closed form. Our results make heavy use of the Chebyshev polynomials of the second kind.