Basic properties of generalized down–up algebras

We introduce a large class of infinite dimensional associative algebras which generalize down–up algebras. Let K be a field and fix f K[x] and r,s,γ K. Define L=L(f,r,s,γ) to be the algebra generated by d,u and h with defining relations: [d,h]r+γd=0, [h,u]r+γu=0, [d,u]s+f(h)=0. Included in this family are Smithʹs class of algebras similar to U(sl2), Le Bruynʹs conformal sl2 enveloping algebras and the algebras studied by Rueda. The algebras L have Gelfand–Kirillov dimension 3 and are Noetherian domains if and only if rs≠0. We calculate the global dimension of L and, for rs≠0, classify the simple weight modules for L, including all finite dimensional simple modules. Simple weight modules need not be classical highest weight modules.