Cyclic operads and algebra of chord diagrams

Abstract

We prove that the algebra $ \cal A $ of chord diagrams, the dual to the associated graded algebra of Vassiliev knot invariants, is isomorphic to the universal enveloping algebra of a Casimir Lie algebra in a certain tensor category (the PROP for Casimir Lie algebras). This puts on a firm ground a known statement that the algebra $ \cal A $ looks and behaves like a universal enveloping algebra. An immediate corollary of our result is the conjecture of [BGRT] on the Kirillov-Duflo isomorphism for algebras of chord diagrams.¶ Our main tool is a general construction of a functor from the category $ \tt CycOp $ of cyclic operads to the category $ \tt ModOp $ of modular operads which is left adjoint to the tree part functor $ {\tt ModOp} \to {\tt CycOp} $ . The algebra of chord diagrams arises when this construction is applied to the operad $ {\tt LIE} $ . Another example of this construction is Kontsevich's graph complex which corresponds to the operad $ {\tt LIE}_\infty $ for homotopy Lie algebras.