Olivier Mathieu
University Lyon I
A global version of Grozman Theorem.
Abstract: Let WX be the Lie algebra of vector elds on a manifold X. Around 1985, Pavel Grozman has classied all WX-equivariant bilinear operators : M N ! P, where M, N and P are tensor density modules, with the additional condition that is a di erential operator. The most interesting case arises when X is of dimension one. In a joint work with K.Iohara, we investigate the same question, without the additional condition that is a di erential operator (since di erential operators are local, we investigate a "global" version of Grozman paper). We give a complete answer when the manifold X is the "circle", i.e. the a ne variety Spec C[z; z^{-1}]. A computer is required to study the determinant of a 6x6 matrix which depends polynomialy on two variables and three parame- ters, It is a key point in the classication of a certain class of Z^N-graded Lie algebras.