Bidiagonal Triples

TL;DR

This paper introduces bidiagonal triples, a new linear algebraic structure extending bidiagonal pairs, and explores their classification and connections to Lie algebra and quantum algebra representations.

Contribution

It generalizes bidiagonal pairs to triples, classifies them using parameter arrays, and links them to $sl_2$ and $U_q(sl_2)$ representation theory.

Findings

Every bidiagonal pair extends uniquely to a bidiagonal triple.

Bidiagonal triples are classified by parameter arrays.

They are closely related to $sl_2$ and $U_q(sl_2)$ representations.

Abstract

We introduce a linear algebraic object called a bidiagonal triple. A bidiagonal triple consists of three diagonalizable linear transformations on a finite-dimensional vector space, each of which acts in a bidiagonal fashion on the eigenspaces of the other two. The concept of bidiagonal triple is a generalization of the previously studied and similarly defined concept of bidiagonal pair. We show that every bidiagonal pair extends to a bidiagonal triple, and we describe the sense in which this extension is unique. In addition we generalize a number of theorems about bidiagonal pairs to the case of bidiagonal triples. In particular we use the concept of a parameter array to classify bidiagonal triples up to isomorphism. We also describe the close relationship between bidiagonal triples and the representation theory of the Lie algebra $sl_2$ and the quantum algebra $U_q(sl_2)$.