The Askey--Wilson algebras, the Lie algebra ${\mathfrak{s}\mathfrak{o}}_3$, and their fermionic realizations

This paper establishes a comprehensive algebraic framework linking the Lie algebra ${\mathfrak{s}\mathfrak{o}}_3$ to the Askey–Wilson algebras. First, we provide a manifestly symmetric reformulation of the algebra homomorphism from the universal Racah algebra $\mathrm{\Re }$ to $U({\mathfrak{s}\mathfrak{l}}_2)$ by exploiting a Lie algebra isomorphism between ${\mathfrak{s}\mathfrak{l}}_2$ and ${\mathfrak{s}\mathfrak{o}}_3$. This perspective facilitates a natural extension to the quantum setting, where we construct an explicit algebra homomorphism from the universal Askey–Wilson algebra ${\mathrm{△}}_{q^4}$ to the nonstandard quantum algebra $U_q^{\mathrm{\prime }}({\mathfrak{s}\mathfrak{o}}_3)$. By viewing the finite-dimensional irreducible $U_q^{\mathrm{\prime }}({\mathfrak{s}\mathfrak{o}}_3)$-modules of classical type as ${\mathrm{△}}_{q^4}$-modules, we demonstrate that the decomposition patterns perfectly parallel the branching rules of $U({\mathfrak{s}\mathfrak{o}}_3)$ over $\mathrm{\Re }$. Furthermore, we extend this correspondence to the fermionic setting by establishing algebra isomorphisms between the skew group rings over $U({\mathfrak{s}\mathfrak{o}}_3)$ and $U_q^{\mathrm{\prime }}({\mathfrak{s}\mathfrak{o}}_3)$ 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.