Temperley-Lieb, Birman-Murakami-Wenzl and Askey-Wilson algebras and other centralizers of $U_q(\mathfrak{sl}_2)$

TL;DR

This paper explores the structure of centralizers of $U_q( ext{sl}_2)$ in tensor products, proposing a conjecture that these centralizers are quotients of the Askey-Wilson algebra, supported by representation analysis.

Contribution

It introduces a conjecture linking centralizers of $U_q( ext{sl}_2)$ to quotients of the Askey-Wilson algebra and verifies it in specific cases.

Findings

Temperley-Lieb, Birman-Murakami-Wenzl, and one-boundary Temperley-Lieb algebras are quotients of the Askey-Wilson algebra.

The conjecture holds in several examined cases.

Provides a generator and relation description of the centralizers.

Abstract

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.