  Ideas of Rozansky and Witten, as developed by Kapranov, show that a complex
symplectic manifold X gives rise to Vassiliev weight systems. In this paper we
study these weight systems by using D(X), the derived category of coherent
sheaves on X. The main idea (stated here a little imprecisely) is that D(X) is
the category of modules over the shifted tangent sheaf, which is a Lie algebra
object in D(X); the weight systems then arise from this Lie algebra in a
standard way. The other main results are a description of the symmetric
algebra, universal enveloping algebra, and Duflo isomorphism in this context,
and the fact that a slight modification of D(X) has the structure of a braided
ribbon category, which gives another way to look at the associated invariants
of links. Our original motivation for this work was to try to gain insight into
the Jacobi diagram algebras used in Vassiliev theory by looking at them in a
new light, but there are other potential applications, in particular to the
rigorous construction of the (1+1+1)-dimensional Rozansky-Witten TQFT, and to
hyperkaehler geometry.
