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

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.