The Askey--Wilson algebras, the Lie algebra $\mathfrak{so}_{3}$, and their fermionic realizations

TL;DR

This paper links the Askey--Wilson algebras with the Lie algebra fso_3, providing symmetric reformulations, quantum extensions, and fermionic analogues to unify integrable algebraic structures and quantum groups.

Contribution

It introduces a symmetric algebraic framework connecting fso_3, Askey--Wilson algebras, and their quantum and fermionic versions, expanding understanding of their interrelations.

Findings

Constructed an explicit homomorphism from ftriangle_{q^4} to U_q'(fso_3)

Demonstrated module decomposition patterns match branching rules of fso_3

Established algebra isomorphisms between skew group rings and anticommutator spin algebras

Abstract

This paper establishes a comprehensive algebraic framework linking the Lie algebra $\mathfrak{so}_{3}$ to the Askey--Wilson algebras. First, we provide a manifestly symmetric reformulation of the algebra homomorphism from the universal Racah algebra $\Re$ to $U(\mathfrak{sl}_2)$ by exploiting a Lie algebra isomorphism between $\mathfrak{sl}_{2}$ and $\mathfrak{so}_{3}$. This perspective facilitates a natural extension to the quantum setting, where we construct an explicit algebra homomorphism from the universal Askey--Wilson algebra $\triangle_{q^4}$ to the nonstandard quantum algebra $U_{q}^{\prime}(\mathfrak{so}_{3})$. By viewing the finite-dimensional irreducible $U_{q}^{\prime}(\mathfrak{so}_{3})$-modules of classical type as $\triangle_{q^4}$-modules, we demonstrate that the decomposition patterns perfectly parallel the branching rules of $U(\mathfrak{so}_3)$ over $\Re$. Furthermore, we extend this correspondence to the fermionic setting by establishing algebra isomorphisms between the skew group rings over $U(\mathfrak{so}_3)$ and $U_q'(\mathfrak{so}_3)$ and their associated anticommutator spin algebras. Collectively, these results provide a unified correspondence that bridges the gap between integrable algebraic structures, quantum groups, and their fermionic analogues.