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 Bar-Natan, Garoufalidis, Rozansky, and Thurston on the Kirillov-Duflo isomorphism for algebras of chord diagrams. Our main tool is a general construction of a functor from the category $\mathtt{CycOp}$ of cyclic operads to the category $\mathtt{ModOp}$ of modular operads which is left adjoint to the ``tree part'' functor $\mathtt{ModOp} \to \mathtt{CycOp}$. The algebra of chord diagrams arises when this construction is applied to the operad for Lie algebras. Another example of this construction is Kontsevich's graph complex that corresponds to the operad for homotopy Lie algebras.