Algebraic structures on modules of diagrams

There exists a graded algebra Λ acting in a natural way on many modules of 3valent diagrams. Every simple Lie superalgebra with a nonsingular invariant bilinear form induces a character on Λ. The classical and exceptional Lie algebras and the Lie superalgebra D(2, 1, α) produce eight distinct characters on Λ and eight distinct families of weight functions on chord diagrams. As a consequence we prove that weight functions coming from semisimple Lie superalgebras do not detect every element in the module A of chord diagrams. A precise description of Λ is conjectured. Introduction. V. Vassiliev [Va] has recently defined a new family of knot invariants. Actually every knot invariant with values in an abelian group may be seen as a linear map from the free Z-module Z[K] generated by isomorphism classes of knots. This module is a Hopf algebra and has a natural filtration Z[K] = I0 ⊃ I1 ⊃ . . . defined in terms of singular knots, and a Vassiliev invariant of order n is an invariant which is trivial on In+1. The coefficients of Jones [J], HOMFLY [H], Kauffman [Ka] polynomials are Vassiliev invariants. The associated graded Hopf algebra GrZ[K] =⊕ n In/In+1 is finitely generated over Z in each degree but its rank is completely unknown. Actually GrZ[K] is a certain quotient of the graded Hopf algebra A of chord diagrams [BN]. Every Vassiliev invariants of order n induces a weight function of degree n, (i.e. a linear form of degree n on A). Conversely every weight function can be integrated (via the Kontsevich integral) to a knot invariant. Very few things are known about the algebra A. Rationally, A is the symmetric algebra on a graded module P, and the so-called Adams operations split A and P in a direct sum of modules defined in terms of unitrivalent diagrams. The rank of P is known in degree < 10. Every Lie algebra equipped with a nonsingular invariant bilinear form and a finite dimensional representation induces a weight function onA. It was conjectured in [BN] Université Paris 7, Institut mathématiques de Jussieu (UMR 7586), 2 place Jussieu 75251 Paris Cedex 05 – Email: [email protected]