| Abstrakt: |
Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a functorZWB ! yA, where B is the category of bottom tangles in handlebodies and yA is the degree-completion of the category A of Jacobi diagrams in handlebodies. As a symmetric monoidal linear category, Ais the linear PROP governing "Casimir Hopf algebras", which are cocommutative Hopf algebras equipped with a primitive invariant symmetric 2-tensor. The functor Z induces a canonical isomorphism grB Š A, where grB is the associated graded of the Vassiliev-Goussarov filtration on B. To each Drinfeld associator ' we associate a ribbon quasi-Hopf algebra H' in yA, and we prove that the braided Hopf algebra resulting from H' by "transmutation" is precisely the image by Z of a canonical Hopf algebra in the braided category B. Finally, we explain how Z refines the LMO functor, which is a TQFT-like functor extending the Le-Murakami-Ohtsuki invariant. [ABSTRACT FROM AUTHOR] |
