The Clebsch–Gordan coefficients of U(sl2) and the Terwilliger algebras of Johnson graphs

Abstract

The universal enveloping algebra $U\left({\mathrm{sl}}_2\right)$ of ${\mathrm{sl}}_2$ is a unital associative algebra over $C$ generated by $E,F,H$ subject to the relations$[H,E]=2E,[H,F]=-2F,[E,F]=H.$ The element$\Lambda =EF+FE+\frac{H^2}{2}$ is called the Casimir element of $U\left({\mathrm{sl}}_2\right)$. Let $\Delta :U\left({\mathrm{sl}}_2\right)\to U\left({\mathrm{sl}}_2\right)\otimes U\left({\mathrm{sl}}_2\right)$ denote the comultiplication of $U\left({\mathrm{sl}}_2\right)$. The universal Hahn algebra $H$ is a unital associative algebra over $C$ 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 $H$. Inspired by the Clebsch–Gordan coefficients of $U\left({\mathrm{sl}}_2\right)$, we discover an algebra homomorphism $♮:H\to U\left({\mathrm{sl}}_2\right)\otimes U\left({\mathrm{sl}}_2\right)$ that maps$A\mapsto \frac{H\otimes 1-1\otimes H}{4},B\mapsto \frac{\Delta \left(\Lambda \right)}{2},C\mapsto E\otimes F-F\otimes E.$ By pulling back via ♮ any $U\left({\mathrm{sl}}_2\right)\otimes U\left({\mathrm{sl}}_2\right)$-module can be considered as an $H$-module. For any integer $n\ge 0$ there exists a unique $(n+1)$-dimensional irreducible $U\left({\mathrm{sl}}_2\right)$-module $L_n$ up to isomorphism. We study the decomposition of the $H$-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.