Primitivity of Noetherian Down-Up Algebras

These algebras arose in the study of the representations of differential partially ordered sets. Examples of down-up algebras include enveloping algebras of many three dimensional Lie algebras and their quantizations. Any down-up algebra A(&, 0, y) with not both a and ,d equal to 0 is isomorphic to a Witten 7-parameter deformation of U ( d 2 ) (see [B], [Wl], [W2]). We assume throughout that A is Noetherian. In [KMP] it was shown that A is Noetherian, if and only if /3 # 0, if and only if A is a domain, and if and only if C[du,ud] is a conlmutative polynomial ring in two indeterminants. As in [BR] consider the roots rl and 7-2 of the polynomial equation x2 a x ,d = 0. Since P = -rlrz, we may assume that ri # 0 for i = 1 , 2 . Answering a question of Jacobson, Ooms [0] gave conditions on a finite dimensional Lie algebra C over a field k of characteristic zero such that its