The Clebsch–Gordan coefficients of [formula omitted] and the Terwilliger algebras of Johnson graphs

Abstract

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 elementΛ=EF+FE+H22 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 mapsA↦H⊗1−1⊗H4,B↦Δ(Λ)2,C↦E⊗F−F⊗E. 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 Ln up to isomorphism. We study the decomposition of the H-module Lm⊗Ln 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.