Non-holonomic modules over Weyl algebras and enveloping algebras

Abstract

A number of counterexamples in the theory of enveloping algebras and Noetherian rings are presented. In particular, we construct non-holonomic, simple modules over the Weyl algebra A, and the enveloping algebra U(S1 z xS12). Further, we show that U(S1 z x S12) is not weakly ideal invariant and that there exist simple S1 z x S12-modules M and E, with E finite dimensional, such that E | is not Artinian. If M is a module over a ring R, then the Gelfand-Kirillov dimension of M, written GK dim M, is defined in [7]. Suppose that M is a simple module over a Noetherian ring R and that G K d i m R < oc. Then M is called holonomic if G K d i m M = 8 9 The reason for the definition is that in the case when R is a Weyl algebra or the enveloping algebra of a finite dimensional, algebraic Lie algebra, holonomic modules have particularly nice properties.