Lie algebras of derivations and affine algebraic geometry over fields of characteristic 0

Abstract

In their article ([15]) Pursell and Shanks have formulated the problem to characterize a manifold by its Lie algebra of all vector fields for the first time. Since then this question has been answered for numerous, different geometric objects: for embedded germs of quasihomogeneous isolated singularities by H. Omori ([13]), for differential, analytic and Stein manifolds by J. Grabowski ([3]) and for germs of arbitrary singularities of dimension at least 3 by H. Hauser and G. MOiler ([6]). Similar considerations for certain abstract algebras have been made by S.M. Skryabin ([16]) and D.A. Jordan ([8]). The purpose of this article is the study of Lie algebras of derivations on finitely generated commutative k-algebras A with no zero divisor, where k is an algebraically closed field of characteristic 0. Though it is also of general interest to study this class of infinite-dimensional Lie algebras in the algebraic case (see [9] Introduction 0.2), there is still another motivation to consider these types of Lie algebras. The investigation of finite--dimensional Lie algebras: