Inspired by the theory of P -and Q-polynomial association schemes we consider the following situation in linear algebra. Let F denote a field, and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A : V → V and A * : V → V satisfying the following four conditions. θ * i−1 − θ * i both equal β + 1, for 2 ≤ i ≤ d − 1. We hope these results will ultimately lead to a complete classification of the TD pairs.

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Let K denote an algebraically closed field and let q denote a nonzero scalar in K that is not a root of unity. Let V denote a vector space over K with finite positive dimension and let A, A * denote a tridiagonal pair on V . Let θ 0 , θ 1 , . . . , θ d (resp. θ * 0 , θ * 1 , . . . , θ * d ) denote a standard ordering of the eigenvalues of A (resp. A * ). We assume there exist nonzero scalars a, a * in K such that θ i = aq 2i−d and θ * i = a * q d−2i for 0 ≤ i ≤ d. We display two irreducible U q ( sl 2 )-module structures on V and discuss how these are related to the actions of A and A * .

Recently B. Hartwig and the second author found a presentation for the three-point sl 2 loop algebra via generators and relations. To obtain this presentation they defined an algebra ⊠ by generators and relations, and displayed an isomorphism from ⊠ to the three-point sl 2 loop algebra. We introduce a quantum analog of ⊠ which we call ⊠ q . We define ⊠ q via generators and relations. We show how ⊠ q is related to the quantum group U q (sl 2 ), the U q (sl 2 ) loop algebra, and the positive part of U q ( sl 2 ). We describe the finite dimensional irreducible ⊠ q -modules under the assumption that q is not a root of 1, and the underlying field is algebraically closed.

Abstract. Motivated by investigations of the tridiagonal pairs of linear transformations, we introduce the augmented tridiagonal algebra T q . This is an infinite-dimensional associative C-algebra with 1. We classify the finite-dimensional irreducible representations of T q . All such representations are explicitly constructed via embeddings of T q into the U q (sl 2 )-loop algebra. As an application, tridiagonal pairs over C are classified in the case where q is not a root of unity.

Let K denote an algebraically closed field with characteristic 0, and let q denote a nonzero scalar in K that is not a root of unity. Let A q denote the unital associative K-algebra defined by generators x, y and relations. We classify up to isomorphism the finite-dimensional irreducible A q -modules on which neither of x, y is nilpotent. We discuss how these modules are related to tridiagonal pairs. Remark 1.2 The equations (2), (3) are the cubic q-Serre relations [32, p. 11].We are interested in a certain class of A q -modules. To describe this class we recall a concept. Let V denote a finite-dimensional vector space over K. A linear transformation X : V → V is said to be nilpotent whenever there exists a positive integer n such that X n = 0. Definition 1.3 Let V denote a finite-dimensional A q -module. We say this module is NonNil whenever the standard generators x, y are not nilpotent on V .

Let K denote an algebraically closed ÿeld with characteristic 0. Let V denote a vector space over K with ÿnite positive dimension and let A; A * denote a tridiagonal pair on V . We make an assumption about this pair. Let q denote a nonzero scalar in K that is not a root of unity. We assume A and A * satisfy the q-Serre relationswhere [3] = (q 3 − q −3 )=(q − q −1 ). Let ( 0; 1; : : : ; d ) denote the shape vector for A; A * . We show the entries in this shape vector are bounded above by binomial coe cients as follows:We obtain this result by displaying a spanning set for V .

Let Γ denote a distance-regular graph with classical parameters (D, b, α, β) and b = 1, α = b − 1. The condition on α implies that Γ is formally self-dual. For b = q 2 we use the adjacency matrix and dual adjacency matrix to obtain an action of the q-tetrahedron algebra ⊠ q on the standard module of Γ. We describe four algebra homomorphisms into ⊠ q from the quantum affine algebra U q ( sl 2 ); using these we pull back the above ⊠ q -action to obtain four actions of U q ( sl 2 ) on the standard module of Γ.