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 $♮:\mathcal{H}\to U({\mathfrak{sl}}_2)\otimes U({\mathfrak{sl}}_2)$ that maps \begin{eqnarray*} A &\mapsto & \frac{H\otimes 1-1\otimes H}{4}, \\ B &\mapsto & \frac{Δ(Λ)}{2}, \\ C &\mapsto & E\otimes F-F\otimes E. \end{eqnarray*} 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.

21 pages

References in corpus (1)

• Entanglement of Free Fermions on Johnson Graphs

Cited by in corpus (1)

• An imperceptible connection between the Clebsch--Gordan coefficients of $U_q({\mathfrak{sl}}_2)$ and the Terwilliger algebras of Grassmann graphs