Cyclic operads and algebra of chord diagrams

Vladimir Hinich, Arkady Vaintrob

Research output: Contribution to journalArticlepeer-review


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 Cyc0p of cyclic operads to the category Mod0p of modular operads which is left adjoint to the "tree part" functor Mod0p → Cyc0p. 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.

Original languageEnglish
Pages (from-to)237-282
Number of pages46
JournalSelecta Mathematica, New Series
Issue number2
StatePublished - 2002


  • Chord diagrams
  • Cyclic operad
  • Modular operad
  • Vassiliev invariants

ASJC Scopus subject areas

  • Mathematics (all)
  • Physics and Astronomy (all)


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