An algebra behind the Clebsch–Gordan coefficients of [formula omitted

Abstract

Let F denote a field. Fix a nonzero q∈F with q4≠1. Let Hq denote a unital associative F-algebra generated by A, B, C and the relations assert that each ofqBC−q−1CBq2−q−2+A,qCA−q−1AC,qAB−q−1BAq2−q−2+C commutes with A, B, C. We call Hq the universal q-Hahn algebra. Motivated by the Clebsch–Gordan coefficients of Uq(sl2), we find a homomorphism ♭:Hq→Uq(sl2)⊗Uq(sl2). We show that the kernel of ♭ is an ideal of Hq generated by a central element of Hq. The decomposition formulae for the tensor products of irreducible Verma Uq(sl2)-modules and of finite-dimensional irreducible Uq(sl2)-modules into the direct sums of finite-dimensional irreducible Hq-modules are also given in the paper.