The Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$ and the Terwilliger algebras of Johnson graphs

Abstract

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 \begin{align*} [H,E]=2E, \qquad [H,F]=-2F, \qquad [E,F]=H. \end{align*} The element $$ \Lambda=EF+FE+\frac{H^2}{2} $$ is called the Casimir element of $U(\mathfrak{sl}_2)$. Let $\Delta:U(\mathfrak{sl}_2)\to U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$ 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 \begin{align*} [C,A]+2A^2+B, \qquad [B,C]+4BA+2C \end{align*} is central in $\mathcal H$. Inspired by the Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$, we discover an algebra homomorphism $\natural:\mathcal H\to U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$ that maps \begin{eqnarray*} A &\mapsto & \frac{H\otimes 1-1\otimes H}{4}, \\ B &\mapsto & \frac{\Delta(\Lambda)}{2}, \\ C &\mapsto & E\otimes F-F\otimes E. \end{eqnarray*} By pulling back via $\natural$ any $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module can be considered as an $\mathcal H$-module. For any integer $n\geq 0$ 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 $m,n\geq 0$. 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.