A Crossed Module giving the Godbillon-Vey Cocycle

We exhibit crossed modules (i.e. certain 4 term exact sequences) corresponding to the Godbillon-Vey generator for W1, V ect(S ) and V ect1,0(Σ). Introduction There is a well known correspondence in homological algebra between (equivalence classes of) exact sequences Λ-modules, starting in M and ending in N with nmodules in between, and elements of ExtΛ(N,M). For Λ = U(g) the universal enveloping algebra of a Lie algebra g, this gives for example a correspondence between H(g, V ) and short eaxct sequences 0 → V → ĝ = V × g → g → 0 where ĝ is the semi-direct product of the abelian Lie algebra V and g. The Lie algebra law on ĝ is given by a 2-cocycle of g with values in V . Note that the short exact sequence in uniquely determined on the vector space level by V and g. In the same way, there is a correspondence between H(g, V ) and certain 4 term exact sequences called crossed modules. In these sequences, only the first and the last term are specified, leaving (du to exactness) a choice of one g-module to complete the sequence. In this article, we exhibit crossed modules corresponding to the GodbillonVey 3-cocycle for W1, V ect(S ) and the 2-dimensional analogue of V ect(S) V ect0,1(Σ) (or V ect1,0(Σ)), Σ being a compact Riemann surface. Acknoledgements The author thanks J.-L. Loday for his question stimulating the present article.