The Terwilliger algebras of Johnson graphs

Let J(n,m)J(n,m) denote the Johnson graph with vertex set X  . Fix a vertex x∈Xx∈X. Let T=T(x)T=T(x) denote the Terwilliger algebra of J(n,m)J(n,m) corresponding to x. In this paper we study the structure of T   under the assumption that m⩾3m⩾3 and n⩾4mn⩾4m. Let U(sl2)U(sl2) be the universal enveloping algebra of sl2sl2. We display a CC-algebra homomorphism ϑ:U(sl2)→Tϑ:U(sl2)→T. We show that T is generated by the image of ϑ and some central elements of T. We obtain these results using the theory of Leonard pairs.