The Terwilliger Algebras Of Johnson Graphs And The Construction Of The Leonard Triples

Terwilliger algebras of association schemes and distance regular graphs are important problems in algebraic combinatorics. In this paper, we give the structure of Terwilliger algebras of Johnson graph using the theory of Leonard pairs and related quantum group-s. Leonard pairs and Leonard triples provide the new algebraic methods for studying distance regular graphs. For the given Leonard pairs of Bannai/Ito type, we construct Leonard triples.The conclusions are as follows.1. Let J(n,m) denote the Johnson graph with vertex set X. Fix a vertex x∈X.. Let T=T(x) denote the Terwilliger algebra of J(n, m) corresponding to x. We study the structure of T under the assumption that m≥3and n≥4m. Let U(sl2) be the universal enveloping algebra of sl2.First, we get the central elements of T. Then we display a C-algebra homomorphism d:U(sl2)â†'T. Finally, we show that T is generated by the image of Ï… and some central elements of T.2. Let K denote an algebraically closed field of characteristic zero and let d denote an even at least3. Let be d+1by d+1matrices. Then A, A*is a Leonard pair on Kd+1of Bannai/Ito type, we determine all the matrices Aε such that A, A*, Aε form a Leonard triple on Kd+1.