The classification of Leonard triples of Racah type

Let K$\mathbb{K}$ denote an algebraically closed field of characteristic zero. Let V   denote a vector space over K$\mathbb{K}$ with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear transformations A  , A^?$A^{?}$, A^ε$A^{\epsilon }$ in End(V)$\mathrm{End}(V)$ such that for each B∈{A,A^?,A^ε}$B\in \{A,A^{?},A^{\epsilon }\}$ there exists a basis for V with respect to which the matrix representing B   is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal. In this paper we define a family of Leonard triples said to have Racah type and classify them up to isomorphism. Moreover, we show that each of them satisfies the _Z3${\mathbb{Z}}_3$-symmetric Askey–Wilson relations. As an application, we construct all Leonard triples that have Racah type from the universal enveloping algebra U(_sl2)$U({\mathit{sl}}_2)$.