# Revisiting the Askey--Wilson algebra with the universal R-matrix of $U_q(sl(2))$

## TL;DR

This paper explores the embedding of the universal Askey--Wilson algebra into the triple tensor product of $U_q(sl_2)$ using the universal R-matrix, revealing new algebraic relations and structures.

## Contribution

It provides a new description of the embedding of AW(3) in $U_q(sl_2)^{ ensor 3}$ via the universal R-matrix, connecting centralizer elements with Casimir conjugations.

## Key findings

- Centralizer elements expressed through R-matrix conjugations
- AW(3) generators identified within $U_q(sl_2)^{ ensor 3}$
- New coaction constructed using the R-matrix

## Abstract

A description of the embedding of the universal Askey--Wilson algebra, AW(3), in $U_q(sl_2)^{\otimes 3}$ is given in terms of the universal R-matrix of $U_q(sl_2)$. The generators of the centralizer of $U_q(sl_2)$ in its three-fold product are naturally expressed through conjugations of Casimir elements with R. They are identified as the images of the generators of AW(3) under the embedding map by showing that they obey the AW(3) relations. This is achieved by introducing a natural coaction also constructed with the help of the R-matrix.