A Connection Behind the Terwilliger Algebras of $H(D,2)$ and $\frac{1}{2}H(D,2)$

The universal enveloping algebra $U({\mathfrak{s}\mathfrak{l}}_2)$ of ${\mathfrak{s}\mathfrak{l}}_2$ is a unital associative algebra over $\mathbb{C}$ generated by $E,F,H$ subject to the relations \begin{align*} [H,E]=2E, \qquad [H,F]=-2F, \qquad [E,F]=H. \end{align*} The distinguished central element $$\mathrm{\Lambda }=EF+FE+\frac{H^2}{2}$$ is called the Casimir element of $U({\mathfrak{s}\mathfrak{l}}_2)$. The universal Hahn algebra $\mathcal{H}$ is a unital associative algebra over $\mathbb{C}$ with generators $A,B,C$ and the relations assert that $[A,B]=C$ and each of \begin{align*} α=[C,A]+2A^2+B, \qquad β=[B,C]+4BA+2C \end{align*} is central in $\mathcal{H}$. The distinguished central element $$\mathrm{\Omega }=4ABA+B^2-C^2-2\beta A+2(1-\alpha )B$$ is called the Casimir element of $\mathcal{H}$. By investigating the relationship between the Terwilliger algebras of the hypercube and its halved graph, we discover the algebra homomorphism $\mathrm{♮}:\mathcal{H}\to U({\mathfrak{s}\mathfrak{l}}_2)$ that sends \begin{eqnarray*} A &\mapsto & \frac{H}{4}, \ B & \mapsto & \frac{E^2+F^2+Λ-1}{4}-\frac{H^2}{8}, \ C & \mapsto & \frac{E^2-F^2}{4}. \end{eqnarray*} We determine the image of $\mathrm{♮}$ and show that the kernel of $\mathrm{♮}$ is the two-sided ideal of $\mathcal{H}$ generated by $\beta$ and $16\mathrm{\Omega }-24\alpha +3$. By pulling back via $\mathrm{♮}$ each $U({\mathfrak{s}\mathfrak{l}}_2)$-module can be regarded as an $\mathcal{H}$-module. For each integer $n\ge 0$ there exists a unique $(n+1)$-dimensional irreducible $U({\mathfrak{s}\mathfrak{l}}_2)$-module $L_n$ up to isomorphism. We show that the $\mathcal{H}$-module $L_n$ ($n\ge 1$) is a direct sum of two non-isomorphic irreducible $\mathcal{H}$-modules.