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

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 ${△}_{q^4}$ 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 ${△}_{q^4}$-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 $U_q^{\prime }({\mathfrak{so}}_3)$ 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.

arXiv comment31 pages