On Leonard Pairs and $q$-Tetrahedron Algebra $\boxtimes_q$

Abstract

Let $\Fa$ denote an algebraically closed field of characteristic zero, fix a nonzero scalar $q\in \Fa$ that is not a root of unity. Consider the $q$-tetrahedron algebra $\boxtimes_q$ over $\Fa$ with standard generators $\{X_{ij}: i, j \in Z_4, j-i=1 \; or \; j-i=2\}$. Let $V$ denote finite dimensional evaluation module for $\boxtimes_q$. In this article for each $r \in Z_4$ and $X_{r+2,r} \in \boxtimes_q$ we find $ A \in \boxtimes_q$ such that the pairs $A, X_{r+2,r}$, $A, X_{r+2,r+3}$, and $A, X_{r+3,r}$ act on $V$ as Leonard pairs. Indeed we will show that $A$ is a linear combination of $X_{r,r+1}$ and $X_{r+1,r+2}$.