An LR pair that can be extended to an LR triple

TL;DR

This paper investigates the conditions under which an LR pair of linear transformations can be extended to form an LR triple, enriching the understanding of their algebraic structure and potential applications.

Contribution

It introduces the concept of extending an LR pair to an LR triple, providing new insights into their structural relationships and potential generalizations.

Findings

Characterization of LR pairs extendable to LR triples

Conditions necessary for extension to an LR triple

Framework for constructing LR triples from LR pairs

Abstract

Fix an integer $d \geq 0$, a field $\mathbb{F}$, and a vector space $V$ over $\mathbb{F}$ with dimension $d+1$. By a decomposition of $V$ we mean a sequence $\{V_i\}_{i=0}^d$ of $1$-dimensional $\mathbb{F}$-subspaces of $V$ such that $V = \sum_{i=0}^d V_i$ (direct sum). Consider $\mathbb{F}$-linear transformations $A$, $B$ from $V$ to $V$. Then $A,B$ is called an LR pair whenever there exists a decomposition $\{V_i\}_{i=0}^d$ of $V$ such that $A V_i = V_{i-1}$ and $B V_i = V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1}=0$ and $V_{d+1}=0$. By an LR triple we mean a $3$-tuple $A,B,C$ of $\mathbb{F}$-linear transformations from $V$ to $V$ such that any two of them form an LR pair. In the present paper, we consider how an LR pair $A,B$ can be extended to an LR triple $A,B,C$.