Bidiagonal pairs, the Lie algebra sl_2, and the quantum group U_q(sl_2)

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 $\SL$ or the quantum group $\uq$. However, its proof makes use of the finite-dimensional representation theory of $\SL$ and $\uq$. In addition to the classification we make explicit the relationship between bidiagonal pairs and modules for $\SL$ and $\uq$.