An embedding of the universal Askey-Wilson algebra into Uq(sl2)⊗Uq(sl2)⊗Uq(sl2)

The Askey-Wilson algebras were used to interpret the algebraic structure hidden in the Racah-Wigner coefficients of the quantum algebra Uq(sl2). In this paper, we display an injection of a universal analog △q of Askey-Wilson algebras into Uq(sl2)⊗Uq(sl2)⊗Uq(sl2) behind the application. Moreover we establish the decomposition rules for 3-fold tensor products of irreducible Verma Uq(sl2)-modules and of finite-dimensional irreducible Uq(sl2)-modules into the direct sums of finite-dimensional irreducible △q-modules. As an application, we derive a formula for the Racah-Wigner coefficients of Uq(sl2).