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

TL;DR

This paper describes the structure of the centralizer of a triple tensor product of quantum group representations using a new algebra that combines Askey-Wilson relations with braid group relations, providing explicit bases.

Contribution

It introduces a quotient of the Askey-Wilson braid algebra that characterizes the centralizer of tensor products of $U_q(sl_2)$ representations, including explicit bases.

Findings

Explicit algebraic presentation of the centralizer.

Construction of a new algebra combining Askey-Wilson and braid relations.

Provision of explicit bases for the centralizer.

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.