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 is an element of K [x] and r, s, gamma is an element of K. Define L = L (f, r, s, gamma) to be the algebra generated by d, u and h with defining relations: [d, h](r) + gammad = 0, [h, u](r) + gammau = 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 not equal 0 . We calculate the global dimension of L and, for rs not equal 0, classify the simple weight modules for L, including all finite dimensional simple modules. Simple weight modules need not be classical highest weight modules. (C) 2004 Elsevier Inc. All rights reserved.