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

The universal enveloping algebra U(sl₂) of sl₂ 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 [Formula presented] is called the Casimir element of U(sl₂). Let Δ:U(sl₂)→U(sl₂)⊗U(sl₂) denote the comultiplication of U(sl₂). 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]+2A²+B,[B,C]+4BA+2C is central in H. Inspired by the Clebsch–Gordan coefficients of U(sl₂), we discover an algebra homomorphism ♮:H→U(sl₂)⊗U(sl₂) that maps [Formula presented] By pulling back via ♮ any U(sl₂)⊗U(sl₂)-module can be considered as an H-module. For any integer n≥0 there exists a unique (n+1)-dimensional irreducible U(sl₂)-module L_n up to isomorphism. We study the decomposition of the H-module L_m⊗L_n for any integers m,n≥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.