Resumen de Cyclic operads and algebra of chord diagrams

V. Hinich, A. Vaintrob

• We prove that the algebra 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 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 CycOp of cyclic operads to the category ModOp of modular operads which is left adjoint to the "tree part" functor ModOp→CycOp . The algebra of chord diagrams arises when this construction is applied to the operad LIE . Another example of this construction is Kontsevich's graph complex which corresponds to the operad LIE∞ for homotopy Lie algebras.