Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of Uq(sl2)

Abstract

Let F denote an algebraically closed field; let q be a nonzero scalar in F such that q is not a root of unity; let d be a nonnegative integer; and let X, Y, Z be the equitable generators of Uq(sl2) over F. Let V denote a finite-dimensional irreducible Uq(sl2)-module with dimension d+1, and let R denote the set of all linear maps from V to itself that act tridiagonally on the standard ordering of the eigenbases for each of X, Y, and Z. We show that R has dimension at most seven. Indeed, we show that the actions of 1, X, Y, Z, XY, YZ, and ZX on V give a basis for R when d≥3.