(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Let denote an algebraically closed field of characteristic zero, fix a nonzero scalar q that is not a root of unity. Consider the q-tetrahedron algebra _q over with standard generators {X_ij: i, j Z_4, j-i=1 or j-i=2}. Let V denote finite dimensional evaluation module for _q. In this article for each r Z_4 and X_r+2,r _q we find A _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.