The universal enveloping algebra $U({\mathfrak{sl}}_2)$ of ${\mathfrak{sl}}_2$ is a unital associative algebra over $\mathbb{C}$ generated by $E,F,H$ subject to the relations$[H,E]=2E,[H,F]=-2F,[E,F]=H.$ The element$\mathrm{\Lambda }=EF+FE+\frac{H^2}{2}$ is called the Casimir element of $U({\mathfrak{sl}}_2)$. Let $\mathrm{\Delta }:U({\mathfrak{sl}}_2)\to U({\mathfrak{sl}}_2)\otimes U({\mathfrak{sl}}_2)$ denote the comultiplication of $U({\mathfrak{sl}}_2)$. The universal Hahn algebra $\mathcal{H}$ is a unital associative algebra over $\mathbb{C}$ generated by $A,B,C$ and the relations assert that $[A,B]=C$ and each of$[C,A]+2A^2+B,[B,C]+4BA+2C$ is central in $\mathcal{H}$. Inspired by the Clebsch–Gordan coefficients of $U({\mathfrak{sl}}_2)$, we discover an algebra homomorphism $♮:\mathcal{H}\to U({\mathfrak{sl}}_2)\otimes U({\mathfrak{sl}}_2)$ that maps$A\mapsto \frac{H\otimes 1-1\otimes H}{4},B\mapsto \frac{\mathrm{\Delta }(\mathrm{\Lambda })}{2},C\mapsto E\otimes F-F\otimes E.$ By pulling back via ♮ any $U({\mathfrak{sl}}_2)\otimes U({\mathfrak{sl}}_2)$-module can be considered as an $\mathcal{H}$-module. For any integer $n\ge 0$ there exists a unique $(n+1)$-dimensional irreducible $U({\mathfrak{sl}}_2)$-module $L_n$ up to isomorphism. We study the decomposition of the $\mathcal{H}$-module $L_m\otimes L_n$ for any integers $m,n\ge 0$. We link these results to the Terwilliger algebras of Johnson graphs. We express the dimensions of the Terwilliger algebras of Johnson graphs in terms of binomial coefficients.

通用包络代数$U（{\mathfrak{斯尔}}_2）$的${\mathfrak{斯尔}}_2$是一个酉结合代数$\mathbb{C}$产生于$乙,F,H$受关系的影响$[H,乙]=2乙,[H,F]=-2F,[乙,F]=H。$元素$\mathrm{\Lambda }=乙F+F乙+\frac{H^2}{2}$称为卡西米尔元素$U（{\mathfrak{斯尔}}_2）$。让$\mathrm{\Delta }：U（{\mathfrak{斯尔}}_2）\to U（{\mathfrak{斯尔}}_2）\otimes U（{\mathfrak{斯尔}}_2）$表示共乘$U（{\mathfrak{斯尔}}_2）$。通用哈恩代数$\mathcal{H}$是一个酉结合代数$\mathbb{C}$产生于$A,乙,C$并且关系断言$[A,乙]=C$和每个$[C,A]+2A^2+乙,[乙,C]+4乙A+2C$是中心于$\mathcal{H}$。受到 Clebsch-Gordan 系数的启发$U（{\mathfrak{斯尔}}_2）$，我们发现代数同态$♮：\mathcal{H}\to U（{\mathfrak{斯尔}}_2）\otimes U（{\mathfrak{斯尔}}_2）$那个地图$A\mapsto \frac{H\otimes 1-1\otimes H}{4},乙\mapsto \frac{\mathrm{\Delta }（\mathrm{\Lambda }）}{2},C\mapsto 乙\otimes F-F\otimes 乙。$通过 ♮ 任意拉回$U（{\mathfrak{斯尔}}_2）\otimes U（{\mathfrak{斯尔}}_2）$- 模块可以被认为是$\mathcal{H}$-模块。对于任意整数$n\ge 0$存在着一个独特的$（n+1）$维数不可约$U（{\mathfrak{斯尔}}_2）$-模块$L_n$直至同构。我们研究了分解$\mathcal{H}$-模块$L_{米}\otimes L_n$对于任意整数$米,n\ge 0$。我们将这些结果与约翰逊图的 Terwilliger 代数联系起来。我们用二项式系数表示约翰逊图的特威利格代数的维数。