An LR pair that can be extended to an LR triple

Abstract

Fix an integer $d\ge 0$, 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 $0\le i\le d$, 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 $A,B,C$.

21 pages

References in corpus (4)

• The quantum algebra $U_q(s{l}_2)$ and its equitable presentation
• Finite-dimensional irreducible $U_q({\mathfrak{sl}}_2)$-modules from the equitable point of view
• Linear transformations that are tridiagonal with respect to the three decompositions for an LR triple
• Lowering-Raising triples and $U_q({\mathfrak{sl}}_2)$