Résumé. Nousétudions des fonctions du paramètre elliptique définies commes intégrales itérées de fonctions elliptiques. Nousétablissons leur lien avec les "associateurs elliptiques" de notre précédent travail au moyen de réalisations fonctionnelles d'algèbres de Lie apparaissant dans cette théorie.

A universal weight function for a quantum affine algebra is a family of functions with values in a quotient of its Borel subalgebra, satisfying certain coalgebraic properties. In representations of the quantum affine algebra it gives off-shell Bethe vectors and is used in the construction of solutions of the qKZ equations. We construct a universal weight function for each untwisted quantum affine algebra, using projections onto the intersection of Borel subalgebras of different types, and study its functional properties.

We construct a genus one analogue of the theory of associators and the Grothendieck-Teichmüller group. The analogue of the Galois action on the profinite braid groups is an action of the arithmetic fundamental group of a moduli space M Q 1, 1 of elliptic curves on the profinite braid groups in genus one. This action factors through an explicit profinite group GT ell , which admits an interpretation in terms of decorations of braided monoidal categories. We relate this group to a prounipotent group scheme GT ell (−). We construct a torsor over this group, the scheme of elliptic associators. An explicit family of elliptic associators is constructed, based on our earlier work with Calaque and Etingof on the universal KZB connexion. The existence of elliptic associators enables one to show that the Lie algebra of GT ell (−) is isomorphic to a graded Lie algebra, on which we obtain several results: semidirect product structure; explicit generators. This existence also allows one to compute the Zariski closure of the mapping class group B 3 in genus one in the automorphism groups of the prounipotent completions of braid groups in genus one. The analytic study of the family of elliptic associators produces relations between MZVs and iterated integrals of Eisenstein series.Dropping associativity constraints and the functor F (which can be put in automatically), the two first conditions mean that the cyclesThe product of these identities yields (14).Each side of (13) with ± = − identifies with the same side of (13) with ± = + and λ ′ replaced by −λ ′ . This implies (13) with ± = −.End of proof of Proposition 2.2. It suffices to prove that (λ ′′ , f ′′ , g ′′ ± ) ∈ GT ell , i.e., that it satisfies conditions (9) and (10).

For a positive integer n we introduce quadratic Lie algebras tr n , qtr n and finitely discrete groups Tr n , QTr n naturally associated with the classical and quantum Yang-Baxter equation, respectively.We prove that the universal enveloping algebras of the Lie algebras tr n , qtr n are Koszul, and compute their Hilbert series. We also compute the cohomology rings for these Lie algebras (which by Koszulity are the quadratic duals of the enveloping algebras). Finally, we construct a basis of U(tr n ).We construct cell complexes which are classifying spaces of the groups Tr n and QTr n , and show that the boundary maps in them are zero, which allows us to compute the integral cohomology of these groups.We show that the Lie algebras tr n , qtr n map onto the associated graded algebras of the Malcev Lie algebras of the groups Tr n , QTr n , respectively. In the case of Tr n , we use quantization theory of 743 Lie bialgebras to show that this map is actually an isomorphism. At the same time, we show that the groups Tr n and QTr n are not formal for n 4.

Hitchin systems, higher Gaudin operators and r-matrices B. Enriquez and V. Rubtsov Abstract. We adapt Hitchin's integrable systems to the case of a punctured curve. In the case of CP 1 and SL n -bundles, they are equivalent to systems studied by Garnier. The corresponding quantum systems were identified by B. Feigin, E. Frenkel and N. Reshetikhin with Gaudin systems. We give a formula for the higher Gaudin operators, using results of R. Goodman and N. Wallach on the center of the enveloping algebras of affine algebras at the critical level. Finally we construct a dynamical r-matrix for Hitchin systems for a punctured elliptic curve, and GL nbundles, and (for n = 2) the corresponding quantum system. Introduction. In [13], N. Hitchin introduced a class of integrable systems, attached to a complex curve X and a semisimple Lie group G. The problem of quantization of these systems was then connected by A. Beilinson and V. Drinfel'd to the Langlands program. This quantization makes use of differential operators on the moduli space of G-bundles on X, obtained from the action of the center of the local completion of the enveloping algebra of a Kac-Moody algebra, at the critical level.This program can also be carried out in the case of curves with marked points. In the particular case of the punctured CP 1 , the action of the center of the enveloping algebra was studied by B. Feigin, E. Frenkel and N. Reshetikhin in [6]; they obtained an integrable system whose first operators are identical to Gaudin's operators ([9]).In this paper, we consider the question of expressing the action of higher central elements. We first note, that the Adler-Kostant-Symes (AKS) scheme, which allows to write families of commuting operators ([2], [14], [21]), can be applied in the present situation, and then show that the higher Hamiltonians obtained in [6], coincide with those. So our problem turns out to be equivalent to expressing higher central elements in the enveloping algebras at critical level, a problem which was solved by several authors ([10], [12]). Here we show how the results of [10] can be used to derive a simple expression of higher Gaudin Hamiltonians.We then turn to the case of punctured elliptic curves. We show that the integrability of Hitchin's system can be deduced from an r-matrix argument. Here r-matrix relations contain additional terms, due to an invariance under the Cartan algebra action. The r-matrix depends on the moduli parameters, so it reminds dynamical r-matrices. In the case of one puncture, our L-operator and r-matrix seem connected with those considered respectively by I. Krichever and A. Gorsky and N. Nekrasov in [15] and [11], and H. Braden, T. Suzuki and E. Sklyanin [5], [19]. It is also analogous to the r-matrix appearing in the work of G. Felder and C. Wieczerkowski on the Knizhnik-Zamolodchikov-Bernard equation on elliptic curves ([7]). We give the form of the first Hamiltonians in this case; one of them contains a Weierstass potential, and so is analogous to the Calogero-Moser Hamiltonian. We comput...

We construct quasi-Hopf algebras quantizing double extensions of the Manin pairs of Drinfeld, associated to a curve with a meromorphic differential, and the Lie algebra sl 2 . This construction makes use of an analysis of the vertex relations for the quantum groups obtained in our earlier work, PBW-type results and computation of R-matrices for them; its key step is a factorization of the twist operator relating "conjugated" versions of these quantum groups.

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Abstract. We construct an algebra morphism from the elliptic quantum group E τ,η (sl 2 ) to a certain elliptic version of the "quantum loop groups in higher genus" studied by V. Rubtsov and the first author. This provides an embedding of E τ,η (sl 2 ) in an algebra "with central extension". In particular we construct L ± -operators obeying a dynamical version of the Reshetikhin-Semenov-TianShansky relations. To do that, we construct the factorization of a certain twist of the quantum loop algebra, that automatically satisfies the "twisted cocycle eqaution" of O. Babelon, D. Bernard and E. Billey, and therefore provides a solution of the dynamical Yang-Baxter equation.Introduction. The aim of this paper is to compare the sl 2 -version of the elliptic quantum groups introduced by the second author ([14]) with quasi-Hopf algebras introduced by V. Rubtsov and the first author ([11, 12]). Elliptic quantum groups are presented by exchange (or "RLL") relations, whereas the algebras of [11] are "quantum loop algebras". Our result can be viewed as an elliptic version of the results of J. Ding and I. Frenkel ([4]) and of S. Khoroshkin ([21]), where Drinfeld's quantum current algebra ([6]) was shown to be isomorphic with the Reshetikhin-Semenov L-operator algebra of [22,13], in the trigonometric and rational case respectively.Elliptic quantum groups are based on a matrix solution R(z, λ) of the dynamical Yang-Baxter equation (YBE). Here "dynamical" means that in addition to the spectral parameter z (belonging to an elliptic curve E), R depends on a parameter λ, which undergo certain shifts in the Yang-Baxter equation. The RLL relations defining the elliptic quantum groups E τ,η (sl 2 ) are then an algebraic variant of the dynamical YBE.In [3], O. Babelon, D. Bernard and E. Billey studied the relation beween the dynamical and quasi-Hopf Yang-Baxter equations. They showed that given a family of twists of a quasi-triangular Hopf algebra, satisfying a certain "twisted cocycle equation", the quasi-Hopf