On a class of finite-dimensional Lie algebras

Abstract

Over fields of prime characteristic the centre of the universal enveloping algebra of a finite-dimensional Lie algebra contains a finitely generated polynomial algebra over which the universal envelope is a finitely generated module. This result, which is due to Curtis, is crucial in certain investigations of finitely generated soluble Lie algebras and motivates the introduction of the class Max-cu, which will be called the class of Curtis algebras, consisting of Lie algebras whose universal envelopes have the property described above. It has been an open question whether the class Max-cu consists of finite-dimensional Lie algebras. This paper gives an affirmative answer to the question.