  I. M. Gelfand and D. B. Fuks have studied the cohomology of the Lie algebra
of vector fields on a manifold. In this article, we generalize their main tools
to compute the Leibniz cohomology, by extending the two spectral sequences
associated to the diagonal and the order filtration. In particular, we
determine some new generators for the diagonal Leibniz cohomology of the Lie
algebra of vector fields on the circle.

K-Theory and Homology