Bidiagonal Triads and the Tetrahedron Algebra

TL;DR

This paper introduces bidiagonal triads, a new linear algebraic structure related to bidiagonal triples, and explores their connection to the representation theory of the tetrahedron Lie algebra.

Contribution

It defines bidiagonal triads, modifies existing theorems from bidiagonal triples, and establishes their relationship with the tetrahedron Lie algebra's representations.

Findings

Bidiagonal triads generalize bidiagonal triples.

Modified theorems from bidiagonal triples to triads.

Established a connection between triads and the tetrahedron Lie algebra.

Abstract

We introduce a linear algebraic object called a bidiagonal triad. A bidiagonal triad is a modification of the previously studied and similarly defined concept of bidiagonal triple. A bidiagonal triad and a bidiagonal triple both consist 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. A triad differs from a triple in the way these bidiagonal actions are defined. We modify a number of theorems about bidiagonal triples to the case of bidiagonal triads. We also describe the close relationship between bidiagonal triads and the representation theory of the tetrahedron Lie algebra.