The centralizer of the image of the diagonal embedding of \(U_q(\mathfrak {sl}_2)\) in the tensor product of three irreducible representations is examined in a Schur–Weyl duality spirit. The aim is to offer a description in terms of generators and relations. A conjecture in this respect is offered with the centralizers presented as quotients of the Askey–Wilson algebra. Support for the conjecture is provided by an examination of the representations of the quotients. The conjecture is also shown to be true in a number of cases thereby exhibiting in particular the Temperley–Lieb, Birman–Murakami–Wenzl and one-boundary Temperley–Lieb algebras as quotients of the Askey–Wilson algebra.

\(U_q(\mathfrak {sl}_2)\)在三个不可约表示的张量积中的对角嵌入图像的中心化器在 Schur-Weyl 对偶精神中进行了检查。目的是提供关于生成器和关系的描述。在这方面的猜想是通过以 Askey-Wilson 代数的商表示的中心化器提供的。对商的表示的检查提供了对猜想的支持。该猜想在许多情况下也被证明是正确的，从而特别展示了 Temperley-Lieb、Birman-Murakami-Wenzl 和单边界 Temperley-Lieb 代数作为 Askey-Wilson 代数的商。