Abstract. We propose a variant to the Etingof-Kazhdan construction of quantization functors. We construct the twistor JΦ associated to an associator Φ using cohomological techniques. We then introduce a criterion ensuring that the “left Hopf algebra ” of a quasitriangular QUE algebra is flat. We prove that this criterion is satisfied at the universal level. This provides a construction of quantization functors, equivalent to the Etingof-Kazhdan construction. Introduction. In [9, 10], Etingof and Kazhdan constructed quantization functors of Lie bialgebras over a field K of characteristic 0. Such a functor is a morphism of props Q: Bialg → ̂S • (̂LBA) with suitable classical limit properties. Here Bialg is the prop of bialgebras and ̂LBA is a completion of the prop of Lie bialgebras. Each quantization functor gives rise to a functor LBA → QUE from the category of Lie bialgebras to that of quantized universal enveloping algebra,

... independently De Concini (unpublished), conjectured that the monodromy of the Casimir connection ∇C introduced in [MTL] is described by Lusztig’s quantum Weyl group operators. This conjecture was proved in [TL1] for all representations of the Lie algebra g = sln and in [TL2] for a number of pairs (g, V) including vector and spin representations of classical Lie algebras and the adjoint representation of all complex, simple Lie algebras. The aim of this paper, and of its sequel [TL4] is to prove this conjecture for all g. Our strategy is inspired by Drinfeld’s proof of the equivalence of the monodromy of the Knizhnik–Zamolodchikov equations for g and the R–matrix representations coming from the quantum group U�g. It relies on the use of quasi–Coxeter algebras, which are to the generalised braid group of type g what Drinfeld’s quasitriangular quasibialgebras are to Artin’s braid groups Bn. Using this notion, and the associated deformation cohomology, which we call Dynkin diagram

Algebra, Pergamon (1970), 329--358. [Dri] V. G. Drinfeld, On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(Q=Q), Algebra i Analiz 2:4 (1990), 149--181. English transl.: Leningrad Math. J. 2 (1991), 829--860. [Kas] C. Kassel, Quantum groups, GTM 155, Springer-Verlag, New York 1995. [KaT] C. Kassel and V. Turaev, Chord diagram invariants of tangles and graphs, Duke Math. J. 92, no. 3 (1998), 497--552. [LM1] T. Q. T. Le and J. Murakami, Kontsevich integral for Kauffman polynomial, preprint Max-Planck-Institut Bonn, 1993. [LM2] T. Q. T. Le and J. Murakami, The universal Vassiliev-Kontsevich invariant for framed oriented links, Comp. Math. 102 (1996), 41--64. [LM3] T. Q. T. Le and J. Murakami, Parallel version of the universal Vassiliev-- Kontsevich invariant, J. Pure and Appl. Algebra 121 (1997), 271--291. [LMO] T. Q. T. Le, J. Murakami and T. Ohtsuki, On a universal perturbative invariant of 3-manifolds, Topology 37-3 (1998), 539--574. [Prz] J. H. Przytyck...

W-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.) make a class of knotted objects which is wider but weaker than their “ordinary” counterparts. To get (say) w-knots from ordinary knots, one has to allow non-planar “virtual” knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation, the “overcrossings commute ” relation, further beyond the ordinary collection of Reidemeister moves, making w-knotted objects a bit weaker once again. The group of w-braids was studied (under the name “welded braids”) by Fenn, Rimanyi and Rourke [FRR] and was shown to be isomorphic to the McCool group [Mc] of “basisconjugating” automorphisms of a free group Fn — the smallest subgroup of Aut(Fn) that contains both braids and permutations. Brendle and Hatcher [BH], in work that traces back to Goldsmith [Gol], have shown this group to be a group of movies of flying rings in R 3. Satoh [Sa] studied several classes of w-knotted objects (under the name “weakly-virtual”) and has shown them to be closely related to certain classes of knotted surfaces in R 4. So w-knotted objects are topologically and algebraically interesting. In this article we study finite type invariants of several classes of w-knotted objects.

We determine explicitly a rational even Drinfeld associator Φ in a completion of the universal enveloping algebra of the Lie superalgebra gl(1|1) ⊕3. More generally, we define a new algebra of trivalent diagrams that has a unique even horizontal group-like Drinfeld associator Φ. The associator Φ is mapped to Φ by a weight system. As a related result of independent interest, we show how O. Viro’s generalization ∆ 1 of the multi-variable Alexander polynomial can be obtained from the universal Vassiliev invariant of trivalent graphs. We determine Φ by using the invariant ∆ 1 ( ) of a planar tetrahedron.

Abstract. We show that any coboundary Lie bialgebra can be quantized. For this, we prove that: (a) Etingof-Kazhdan quantization functors are compatible with Lie bialgebra twists, and (b) if such a quantization functor corresponds to an even associator, then it is also compatible with the operation of taking coopposites. We also use the relation between the Etingof-Kazhdan construction of quantization functors and the alternative approach to this problem, which was established in a previous work. Let k be a field of characteristic 0. Unless specified otherwise, “algebra”, “vector space”, etc., means “algebra over k”, etc.

Abstract A rational Ansatz is proposed for the generating function � j,k β2j+k,2jxjy k, where βm,u is the number of primitive chinese character diagrams with u univalent and 2m − u trivalent vertices. For Pm: = � u≥2 βm,u, the conjecture leads to the sequence

Abstract. This is an overview article on finite type invariants, written for the Encyclopedia of Mathematical

The workshop and seminars on “Invariants of knots and 3-manifolds ” was held at

alors le probleme d'etudier l'algebre engendree par des generateurs (les multizeta symboliques) indexes par les k-uplets s et definie par les relations donnees par ces deux produits de melange; on obtient ainsi l'algebre MZV, dont Ecalle est en train de determiner la structure. Pour definir des series generatrices permettant de coder simultanement tous les multizeta symboliques il convient d'inclure les s correspondant a des series divergentes (avec s 1 = 1). Nous mentionnons enfin brievement quelques resultats de l'equipe de Petitot sur les polylogarithmes, et encore plus brievement plusieurs sujets connexes. http://www.math.jussieu.fr/#miw/articles/ps/MZV.ps 2 1. Le produit de melange lie aux series Pour s et s # entiers #<F14.58