We introduce a linear algebraic object called a bidiagonal pair. Roughly speaking, a bidiagonal pair is a pair of diagonalizable linear transformations on a finite-dimensional vector space, each of which acts in a bidiagonal fashion on the eigenspaces of the other. We associate to each bidiagonal pair a sequence of scalars called a parameter array. Using this concept of a parameter array we present a classification of bidiagonal pairs up to isomorphism. The statement of this classification does not explicitly mention the Lie algebra sl 2 or the quantum group U q (sl 2 ). However, its proof makes use of the finite-dimensional representation theory of sl 2 and U q (sl 2 ). In addition to the classification we make explicit the relationship between bidiagonal pairs and modules for sl 2 and U q (sl 2 ).

Let V denote a finite dimensional vector space over an algebraically closed field. Let U 0 U 1 U d denote a sequence of nonzero subspaces whose direct sum is V . Let R V → V and L V → V denote linear transformations with the following properties: forare bijections; the maps R and L satisfy the cubic q-Serre relations where q is nonzero and not a root of unity. Let K V → V denote the linear transformation such that K − q 2i−d I U i = 0 for 0 ≤ i ≤ d. We show that there exists a unique U q 2 -module structure on V such that each of R − e − 1 , L − e − 0 , K − K 0 , and K −1 − K 1 vanish on V , where e − 1 e − 0 K 0 K 1 are Chevalley generators for U q 2 . We determine which U q 2 -modules arise from our construction.

For any field K and integer n ≥ 2 we consider the Leavitt algebra L K (n); for any integer d ≥ 1 we form the matrix ring S = M d (L K (n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab − ba for a, b ∈ S. We denote this Lie algebra as S − , and consider its Lie subalgebra [S − , S − ]. In our main result, we show that [S − , S − ] is a simple Lie algebra if and only if char(K) divides n − 1 and char(K) does not divide d. In particular, when d = 1 we get that [L K (n) − , L K (n) − ] is a simple Lie algebra if and only if char(K) divides n − 1.2000 Mathematics Subject Classification. Primary 16D30, 16S99, 17B65.

The q-tetrahedron algebra ⊠ q was recently introduced and has been studied in connection with tridiagonal pairs. In this paper we further develop this connection. 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) denote a standard ordering of the eigenvalues of A (resp. A * ). T. Ito and P. Terwilliger have shown that when θ i = q 2i−d and θ * i = q d−2i (0 ≤ i ≤ d) there exists an irreducible ⊠ qmodule structure on V such that the ⊠ q generators x 01 , x 23 act as A, A * respectively. In this paper we examine the case in which there exists a nonzero scalar c in K such that θ i = q 2i−d and θ * i = q 2i−d + cq d−2i for 0 ≤ i ≤ d. In this case we associate to A, A * a polynomial P in one variable and prove the following theorem as our main result. Theorem The following are equivalent:(i) There exists a ⊠ q -module structure on V such that x 01 acts as A and x 30 + cx 23 acts as A * , where x 01 , x 30 , x 23 are standard generators for ⊠ q .(ii) P (q 2d−2 (q − q −1 ) −2 ) = 0.Suppose (i),(ii) hold. Then the ⊠ q -module structure on V is unique and irreducible.

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