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

Abstract

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 $A_q$ denote the unital associative $K$-algebra defined by generators $x,y$ and relations x^3y-[3]_q x^2yx +[3]_q xyx^2 -yx^3=0, y^3x-[3]_q y^2xy +[3]_q yxy^2 -xy^3=0, where $[3]_q = (q^3-q^{-3})/(q-q^{-1})$. We classify up to isomorphism the finite-dimensional irreducible $A_q$-modules on which neither of $x,y$ is nilpotent. We discuss how these modules are related to tridiagonal pairs.