An algebra behind the Clebsch–Gordan coefficients of U_q(sl₂)

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