Relations among the universal Racah algebra, the anticommutator spin algebra and a skew group ring over $U(\mathfrak{sl}_2)$
Hau-Wen Huang
TL;DR
The paper studies how the universal Racah algebra $\Re$ embeds into $U(\mathfrak{sl}_2)$ and how its fermionic analogue, the anticommutator spin algebra $\mathcal{A}$, fits into a skew group ring $U(\mathfrak{sl}_2)_{\mathbb{Z}/2\mathbb{Z}}$. It reformulates the known $\Re\to U(\mathfrak{sl}_2)$ realization in a symmetric way via the isomorphism $\mathfrak{so}_3\simeq \mathfrak{sl}_2$, and constructs an embedding $\mathcal{A}\hookrightarrow U(\mathfrak{sl}_2)_{\mathbb{Z}/2\mathbb{Z}}$ with $J_1\mapsto\frac{(E+F)\rho}{2}$, $J_2\mapsto\frac{H}{2}$, $J_3\mapsto\frac{(E-F)\rho}{2}$, while also obtaining a homomorphism $\Re\to \mathcal{A}$ with $A\mapsto\frac{(J_1-1)(J_1+1)}{4}$, $B\mapsto\frac{(J_2-1)(J_2+1)}{4}$, $C\mapsto\frac{(J_3-1)(J_3+1)}{4}$ and $\alpha,\beta,\gamma\mapsto 0$; the composition of these maps recovers the original $\Re\to U(\mathfrak{sl}_2)$, clarifying the structural relationships among $\Re$, $\mathcal{A}$, and $U(\mathfrak{sl}_2)$. The proofs establish the skew group ring framework, prove the isomorphism $\mathcal{A}_{\mathbb{Z}/2\mathbb{Z}}\cong U(\mathfrak{sl}_2)_{\mathbb{Z}/2\mathbb{Z}}$, and demonstrate the injective embedding $\mathcal{A}\hookrightarrow U(\mathfrak{sl}_2)_{\mathbb{Z}/2\mathbb{Z}}$, together with the explicit realizations of $\Re$ inside $\mathcal{A}$ and the compatibility with $U(\mathfrak{sl}_2)$ via a commutative diagram.
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.