Basic properties of generalized down–up algebras

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