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.