Down–Up Algebras and Their Representation Theory

A class of algebras called down–up algebras was introduced by G. Benkart and T. Roby (1998, J. Algebra209, 305–344). We classify the finite dimensional simple modules over Noetherian down–up algebras and show that in some cases every finite dimensional module is semisimple. We also study the question of when two down–up algebras are isomorphic.