Cyclic operads and algebra of chord diagrams

• V. Hinich ^[1] ; A. Vaintrob ^[2]
  1. [1] University of Haifa

     University of Haifa

     Israel

     
  2. [2] University of Oregon, USA
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 8, Nº. 2, 2002, págs. 237-282
• Idioma: inglés
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• Resumen

  • 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.