It is only in recent times that infinite-dimensional Lie algebras have
been the subject of other than sporadic study, with perhaps two
exceptions: Cartan's simple algebras of infinite type, and free
algebras. However, the last decade has seen a considerable increase of
interest in the subject, along two fronts: the topological and the
algebraic. The former, which deals largely with algebras of operators on
linear spaces, or on manifolds modelled on linear spaces, has been dealt
with elsewhere*). The latter, which is the subject of the present
volume, exploits the surprising depth of analogy which exists between
infinite-dimen- sional Lie algebras and infinite groups. This is not to
say that the theory consists of groups dressed in Lie-algebraic
clothing. One of the tantalising aspects of the analogy, and one which
renders it difficult to formalise, is that it extends to theorems better
than to proofs. There are several cases where a true theorem about
groups translates into a true theorem about Lie algebras, but where the
group-theoretic proof uses methods not available for Lie algebras and
the Lie-theoretic proof uses methods not available for groups. The two
theories tend to differ in fine detail, and extra variations occur in
the Lie algebra case according to the underlying field. Occasionally the
analogy breaks down altogether. And of course there are parts of the Lie
theory with no group-theoretic counterpart.