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 Smith’s class of algebras similar to U(sl2), Le Bruyn’s 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 6= 0. We calculate the global dimension of L and, for rs 6= 0, classify the simple weight modules for L, including all finite dimensional simple modules. Simple weight modules need not be classical highest weight modules.