Bidiagonal Triads and the Tetrahedron 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.

25 pages, submitted for publication to Communications in Algebra