Askey-Wilson braid algebra and centralizer of $U_q(\mathfrak{sl}_2)$

Abstract

A presentation of the centralizer of the three-fold tensor product of the spin $s$ representation of the quantum group $U_q(\mathfrak{sl}_2)$ is provided. It is expressed as a quotient of the Askey-Wilson braid algebra. This newly defined algebra combines the Askey-Wilson relations with the braid group relations, on three strands, together with a characteristic equation of degree $2s+1$ for the braid generators. Explicit bases are given for the centralizer.