• DOI:10.1007/S00029-002-8106-2
• Corpus ID: 17214400

Cyclic operads and algebra of chord diagrams

@article{Hinich2000CyclicOA,
  title={Cyclic operads and algebra of chord diagrams},
  author={Vladimir Hinich and Arkady Vaintrob},
  journal={Selecta Mathematica},
  year={2000},
  volume={8},
  pages={237-282}
}

• V. Hinich, A. Vaintrob
• Published 19 May 2000
• Mathematics
• Selecta Mathematica

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.