Basic properties of generalized down–up algebras

Abstract

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 Smith's class of algebras similar to U( sl 2), Le Bruyn's conformal sl 2 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.