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Let  denote an algebraically closed field with characteristic 0, and let q denote a nonzero scalar in  that is not a root of unity. Let q denote the unital associative -algebra defined by generators x,y and relations \begin{eqnarray*} x^{3}y - [3]_{q} x^{2}yx + [3]_{q} xyx^{2} - yx^{3} &=& 0,\\ y^{3}x - [3]_{q} y^{2}xy + [3]_{q} yxy^{2} - xy^{3} &=& 0, \end{eqnarray*} where [3]q = (q3 - q-3)/(q - q-1). We classify up to isomorphism the finite-dimensional irreducible q-modules on which neither of x,y is nilpotent. We discuss how these modules are related to tridiagonal pairs.

TWO NON-NILPOTENT LINEAR TRANSFORMATIONS THAT SATISFY THE CUBIC q-SERRE RELATIONS