Two non-nilpotent linear transformations that satisfy the cubic q-Serre relations

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Let K denote an algebraically closed field with characteristic 0, and let q denote a nonzero scalar in K that is not a root of unity. Let Aq denote the unital associative K-algebra defined by generators x,y and relations x³y-[3]q x²yx +[3]q xyx² -yx³=0, y³x-[3]q y²xy +[3]q yxy² -xy³=0, where [3]q = (q³-q⁻³)/(q-q⁻¹). We classify up to isomorphism the finite-dimensional irreducible Aq-modules on which neither of x,y is nilpotent. We discuss how these modules are related to tridiagonal pairs.