Relations among the universal Racah algebra, the anticommutator spin algebra and a skew group ring over $U(\mathfrak{sl}_2)$

Abstract: We revisit the algebra homomorphism from the universal Racah algebra $\Re$ into $U(\mathfrak{sl}2)$, originally introduced in connection with representation-theoretic and combinatorial models. Using a Lie algebra isomorphism $\mathfrak{so}_3\to \mathfrak{sl}_2$, we reformulate this homomorphism in a more symmetric form. The anticommutator spin algebra $\mathcal{A}$, which serves as a fermionic counterpart to the standard angular momentum algebra, can be regarded as an anticommutator analogue of $U(\mathfrak{so}{3})$. In analogy with this isomorphism, we embed $\mathcal A$ into a skew group ring of $\mathbb{Z}/2\mathbb{Z}$ over $U(\mathfrak{sl}2)$, denoted by $U(\mathfrak{sl}_2){\mathbb{Z}/2\mathbb{Z}}$. We further construct a realization of $\Re$ within $\mathcal A$, corresponding to the homomorphism $\Re\to U(\mathfrak{so}3)$. Composed with the embedding $\mathcal A \hookrightarrow U(\mathfrak{sl}_2){\mathbb{Z}/2\mathbb{Z}}$, this realization recovers the original homomorphism $\Re \to U(\mathfrak{sl}_2)$ and thereby clarifies the relationships among these algebras.