An embedding of the universal Askey-Wilson algebra into $U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)$

TL;DR

This paper constructs an embedding of the universal Askey-Wilson algebra into a triple tensor product of quantum groups, revealing new decomposition rules for tensor products of modules.

Contribution

It introduces an injection of the universal Askey-Wilson algebra into a tensor product of quantum groups and derives decomposition rules for tensor products of modules.

Findings

Injection of $ riangle_q$ into $U_q(sl_2)^{ ensor 3}$ established.

Decomposition rules for tensor products of Verma modules formulated.

Decomposition of finite-dimensional modules into $ riangle_q$-modules achieved.

Abstract

The Askey--Wilson algebras were used to interpret the algebraic structure hidden in the Racah--Wigner coefficients of the quantum algebra $U_q(\mathfrak{sl}_2)$. In this paper, we display an injection of a universal analog $\triangle_q$ of Askey--Wilson algebras into $U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)$ behind the application. Moreover we formulate the decomposition rules for $3$-fold tensor products of irreducible Verma $U_q(\mathfrak{sl}_2)$-modules and of finite-dimensional irreducible $U_q(\mathfrak{sl}_2)$-modules into the direct sums of finite-dimensional irreducible $\triangle_q$-modules.