Structure theory of finite conformal algebras

In this paper we develop a structure theory of finite rank conformal algebras. Applications of this theory are two-fold. On the one hand, the conformal algebra structure is an axiomatic description of the operator product expansion (OPE) of chiral fields in a conformal field theory. Hence the theory of finite conformal algebras provides a classication of finite systems of fields closed under the OPE. On the other hand, the category of finite conformal algebras is (more or less) equivalent to the category of finite-dimensional Lie algebras spanned by Fourier coefficients of a finite number of pairwise local fields (or rather formal distributions) that are closed under the OPE. Hence the theory of finite conformal algebras provides a classication of these finite "formal distribution Lie algebras".