Let F denote a field, and let V denote a vector space of finite positive dimension over F. Let A, A * denote a tridiagonal pair on V of diameter d 3. Assume the eigenvalue and dual eigenvalue sequences of A, A * satisfy θ i = q 2i θ , θ * i = q 2d−2i θ * for some nonzero scalars θ , θ * , q ∈ F, where q is not a root of unity. Assume that not all eigenvalues of A and A * have multiplicity one. Let M and M * denote the subalgebras of End(V ) generated by A and A * , respectively, and assume that V = Mv * + M * v for some eigenvectors v * of A * associated with θ * 0 and v of A associated with θ d . We find a nice basis for V and describe the action of A, A * on this basis in terms of six parameters.

Abstract.A type of tridiagonal pair is considered, said to be mild of q-Serre type. It is shown that these tridiagonal pairs induce the structure of a quantum affine algebra Uq( sl 2 )-module on their underlying vector space. This is done by presenting an explicit basis for the underlying vector space and describing the Uq( sl 2 )-action on that basis.Key words. Leonard pair, Tridiagonal pair, Quantum affine algebra.AMS subject classifications. 20G42, 15A04, 33D80, 05E35, 33C45, 33D45.1. Introduction. In [1] the authors study the mild tridiagonal pairs of q-Serre type-the main result is a description of the members of this family by their action on an "attractive" basis for the underlying vector space. In this paper we use this action to describe a U q ( sl 2 )-module structure on the underlying vector space of each mild tridiagonal pair of q-Serre type. We do so by constructing linear operators on this vector space which essentially satisfy the defining relations for U q ( sl 2 ) in the Chevalley presentation. To state our result precisely we recall some definitions. Throughout this paper, let F denote a field, and let V denote a vector space over

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Abstract. We construct Leonard pairs from finite-dimensional irreducible sl 2 -modules, using the equitable basis for sl 2 . We show that our construction yields all Leonard pairs of Racah, Hahn, dual Hahn, and Krawtchouk type, and no other types of Leonard pairs.