The present paper was motivated by the problem of classification of operator product expansions (OPE) in conformal field theory. This problem was solved in [DK] in the case when the chiral algebra is generated by finitely many bosonic fields such that in their OPE only linear combinations of these fields and their derivatives occur. An axiomatic description of such a system of fields is called a finite conformal algebra [K6]. The classification of finite conformal algebras uses in an essential way Cartan's classification of pseudogroups of transformations of a finitedimensional manifold, which, in the modern language, is equivalent to the classification, up to formal equivalence, of Lie algebras of vector fields on a finite-dimensional manifold. The problem of classification of OPE when fermionic fields are allowed as well, or, equivalently, of finite conformal superalgebras, requires an extension of Cartan's theory to the case of supermanifolds. Below I explain the problem in more detail.
Elie Cartan published a solution to the problem (posed by Sophus Lie) of classification of simple infinite-dimensional Lie algebras of vector fields on a finite-dimensional manifold in 1909 [C]. This work had been virtually forgotten until the sixties. A resurgence of interest in this area began with the work of Singer and Sternberg [SS] and of Guillemin and Sternberg [GS], which developed an adequate language and machinery of filtered and graded Lie algebras.
The basic problem of the theory is to classify, up to formal equivalence, infinite-dimensional Lie algebras of vector fields acting transitively in a neighborhood of a point x of a complex manifold X. Let L be such a Lie algebra and let L k (k # Z + ) denote the subalgebra of L consisting of vector Article No. AI981756