Askey–Wilson Braid Algebra and Centralizer of $$U_q(\mathfrak {sl}_2)$$

Abstract

A presentation of the centralizer of the threefold 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.