BIDIAGONAL PAIRS, THE LIE ALGEBRA 𝔰𝔩2, AND THE QUANTUM GROUP Uq(𝔰𝔩2)

Abstract

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 𝔰𝔩2 or the quantum group Uq(𝔰𝔩2). However, its proof makes use of the finite-dimensional representation theory of 𝔰𝔩2 and Uq(𝔰𝔩2). In addition to the classification we make explicit the relationship between bidiagonal pairs and modules for 𝔰𝔩2 and Uq(𝔰𝔩2).