Some Leonard Triples, Racah Algebras Based On Distance-regular Graphs Of Racah Type

By a Leonard triple, we mean a triple of diagonalizable operators on a finite-dimensional vector space such that for each operator, there is an ordering of an eigenbasis for the se-lected operator with respect to which the other two operators are irreducible tridiagonal.Let C denote the field of complex numbers and let D denote an integer at least 3. Let 1/2H"(2D+1,2) denote the halved graph of the (2D+1)-cube with respect to the original P-polynomial structure R0, R1,..., RD and another Q-polynomial structure E0, E2, E4,..., E3, E1 in terms of the original ones. Let 1/2(?)(4D,2) denote the folded halved graph of the 4D-cube and let 1/2H(4D+2,2) denote the folded halved graph of the (4D+2)-cube. Note that they are all distance-regular graphs of Racah type.In this paper we consider the relations between the above three graphs and the Leonard triples or the Racah algebra over C Our results are described as follows.1. Fix a vertex of 1/2H"(2D+1,2) and let T1 denote the corresponding Terwilliger algebra with respect to this vertex. We first construct three elements (?)1,(?)1* and (?)1ε of T1. Then we show that the triple (?)1,(?)1*,(?)1ε acts on each irreducible T1-module as a Leonard triple. Moreover, let (?)1 be a Racah algebra with its generators and real parameters satisfying certain conditions. We display a C-algebra homomorphism from (?)1 to T1.2. Fix a vertex of 1/2H(4D,2) and let T2 denote the Terwilliger algebra of 1/2H[4D,2) with respect to this vertex. We construct three elements (?)2, (?)2*,(?)2ε of T2 and show that the triple (?)2, (?)2*, (?)2ε not only acts on each irreducible T2-module as a Leonard triple but also satisfies some very appealing equations. Moreover, let W denote an irreducible T2-module with type φ and let (?)φ be a Racah algebra with respect to φ. Then there exists (?)φ-module structure on W.3. Fix a vertex of 1/2(?)(4D+2,2) and let T3 denote the Terwilliger algebra of 1/2(?)(4D+2,2) with respect to this vertex. We construct three elements (?)3,(?)3*,(?)3ε of T3 and show that the triple (?)3,(?)3*,(?)3ε not only acts on each irreducible T3-module as a Leonard triple but also satisfies some very appealing equations. Moreover, let W denote an irreducible T3-module with auxiliary parameter e and let (?)e be a Racah algebra with respect to e. Then there exists (?)e-module structure on W.