The Leonard triples having classical type

Let KK denote an algebraically closed field of characteristic zero. Let V   denote a vector space over KK with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear transformations A  , A⁎A⁎, AεAε in End(V)End(V) such that for each B∈{A,A⁎,Aε}B∈{A,A⁎,Aε} 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 classical type   and show that these Leonard triples consist of two families: the Racah type and the Krawtchouk type. Moreover, we construct all Leonard triples that have classical type from the universal enveloping algebra U(sl2)U(sl2).