We determine explicitly a rational even Drinfeld associator Q 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 P. The associator P is mapped to Q by a weight system. As a result of independent interest, we show how O. Viro's generalization Delta^1 of the multi-variable Alexander polynomial can be obtained from the universal Vassiliev invariant of trivalent graphs. We determine P by using the invariant Delta^1(T) of a planar tetrahedron T.
We prove that the Drinfeld generators of grt_1(k) span the image of this Lie algebra in the abelianization of the commutator of the free Lie algebra with two generators. We show that this result implies Gamma-function formulas for arbitrary associators.
We construct a family of Drinfeld associators interpolating between the Knizhnik-Zamolodchikov associator, the Alekseev-Torossian associator and the anti-Knizhnik-Zamolodchikov associator. We give explicit integral formulæ for the family of elements of the Grothendieck-Teichmüller Lie algebra tangent to the family of associators. As an application, we settle a conjecture of Pavel Etingof about the Alekseev-Torossian associator. Furthermore, we give explicit integral formulæ for the family of stable formality morphisms corresponding (in a precise way) to the above family of associators, and for…
We prove an estimate on denominators of rational Drinfeld associators. To obtain this result, we prove the corresponding estimate for the p-adic associators stable under the action of suitable elements of Gal(\bar{Q}/Q). As an application, we settle in the positive Duflo's question on the Kashiwara--Vergne factorizations of the Jacobson element J_p(x,y)=(x+y)^p-x^p-y^p in the free Lie algebra over a field of characteristic p. Another application is a new estimate on denominators of the Kontsevich knot invariant.
The Kashiwara-Vergne (KV) conjecture states the existence of solutions of a pair of equations related with the Campbell-Baker-Hausdorff series. It was solved by Meinrenken and the first author over the real numbers, and in a formal version, by the first and last authors over a field of characteristic 0. In this paper, we give a simple and explicit formula for a map from the set of Drinfeld associators to the set of solutions of the formal KV equations. Both sets are torsors under the actions of prounipotent groups, and we show that this map is a morphism of torsors. When specialized to the KZ …
We give explicit formulas for the first few coefficients of the Alekseev-Torossian associator and a second Drinfeld associator. This is done by analyzing the free and transitive action of the Grothendieck-Teichmüller group and its Lie algebra $\mathfrak{grt}_1$ on the set of Drinfeld associators. As a result we obtain that, up to a conjecture on multiple zeta values, both associators are not rational.
Drinfeld associator is a key tool in computing the Kontsevich integral of knots. A Drinfeld associator is a series in two non-commuting variables, satisfying highly complicated algebraic equations - hexagon and pentagon. The logarithm of a Drinfeld associator lives in the Lie algbera L generated by the symbols a,b,c modulo [a,b]=[b,c]=[c,a]. The main result is a description of compressed associators that satisfy the compressed pentagon and hexagon in the quotient L/[[L,L],[L,L]]. The key ingredient is an explicit form of Campbell-Baker-Hausdorff formula in the case when all commutators commute…
We develop a higher genus version of Drinfeld associators by means of operad theory. We start by introducing a framed version of rational associators and Grothendieck-Teichmüller groups and show that their definition is independent of the framing data. Next, we define a framed version of the universal KZ connection and we use it to show that over the complex numbers, the rational framed Drinfeld torsor is not empty. Next, we concentrate on the higher genus version of this story. We define an operad module of framed parenthesized higher genus braidings in prounipotent groupoids and we define it…
The Kashiwara-Vergne (KV) conjecture is a property of the Campbell-Hausdorff series put forward in 1978. It has been settled in the positive by E. Meinrenken and the first author in 2006. In this paper, we study the uniqueness issue for the KV problem. To this end, we introduce a family of infinite dimensional groups KV_n, and an extension \hat{KV}_2 of the group KV_2. We show that the group \hat{KV}_2 contains the Grothendieck-Teichmueller group GRT as a subgroup, and that it acts freely and transitively on the set of solutions of the KV problem Sol(KV). Furthermore, we prove that Sol(KV) is …
We give a generalization of the Reshetikhin-Turaev functor for tangles to get a combinatorial formula for the universal Vassiliev-Kontsevich invariant of framed oriented links which is coincident with the Kontsevich integral. The universal Vassiliev-Kontsevich invariant is constructed using the Drinfeld associator. We prove the uniqueness of the Drinfeld associator. As a corollary one gets the rationality of the Kontsevich integral. Many properties of the universal Vassiliev-Kontsevich invariant are established. Connections to quantum group invariants and to multiple zeta values are discussed.
Recall that Tamarkin's construction arXiv:math/9803025, arXiv:math/0003052 gives us a map from the set of Drinfeld associators to the set of homotopy classes of L-infinity quasi-isomorphisms for Hochschild cochains of a polynomial algebra. Due to results of V. Drinfeld (Algebra i Analiz 2, (1990)) and T. Willwacher arXiv:1009.1654 both the source and the target of this map are equipped with natural actions of the Grothendieck-Teichmueller group $GRT_1$. In this paper, we use the result from arXiv:1305.4699 to prove that this map from the set of Drinfeld associators to the set of homotopy class…
In arXiv:math/0105152, the second author used the Kontsevich deformation quantization technique to define a natural connection ω_n on the compactified configuration spaces of n points on the upper half-plane. This connection takes values in the Lie algebra of derivations of the free Lie algebra with n generators. In this paper, we show that ω_n is flat. The configuration space contains a boundary stratum at infinity which coincides with the (compactified) configuration space of n points on the complex plane. When restricted to this stratum, ω_n gives rise to a flat connection ω_n^\infty. We sh…
We develop a cyclotomic analogue of the theory of associators. Using a trigonometric version of the universal KZ equations, we prove the formality of a morphism B_n^1 -> (ZZ/N ZZ)^n rtimes S_n, where B_n^1 is a braid group of type B. The formality isomorphism depends algebraically on a series Psi_KZ, the ``KZ pseudotwist''. We study the scheme of pseudotwists and show that it is a torsor under a group GTM(N,k), mapping to Drinfeld's group GT(k), and whose Lie algebra is isomorphic to its associated graded grtm(N,k). We prove that Ihara's subgroup GTK of the Grothendieck-Teichmüller group, defi…
Using properties of ordered exponentials and the definition of the Drinfeld associator as a monodromy operator for the Knizhnik-Zamolodchikov equations, we prove that the analytic and the combinatorial definitions of the universal Vassiliev invariants of links are equivalent.
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Institute for Information Transmission Problems（信息传输问题研究所）；NRC "Kurchatov Institute"（库尔恰托夫国家研究中心）
This paper is a continuation of the series of papers "Quantization of Lie bialgebras (QLB) I-V". We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in QLB-I,II is isomorphic to the Drinfeld-Jimbo quantization of this Lie bialgebra, with the standard quasitriangular structure. This implies that when the quantization parameter is formal, then the category O for the quantized Kac-Moody algebra is equivalent, as a braided tensor category, to the category O over the corresponding classical Kac-Moody algebra, with th…
The values at positive integers of the polyzeta functions are solutions of the polynomial equations arising from Drinfeld's associators, which have numerous applications in quantum algebra. Considered as iterated integrals they become periods of the motivic fundamental groupoid of $\mathbb{P}^1\setminus\{0,1,\infty\}$. From there comes a fundamental, yet no more explicit, system of algebraic relations; it implies the system of associators. We focus here on the combinatorics properties of another system of relations, the ``double shuffles'', which comes from elementary series and integrals mani…
The universal Vassiliev-Kontsevich invariant is a functor from the category of tangles to a certain graded category of chord diagrams, compatible with the Vassiliev filtration and whose associated graded is an isomorphism. The Vassiliev filtration has a natural extension to tangles in any thickened surface $M\times I$ but the corresponding category of diagrams lacks some finiteness properties which are essential to the above construction. We suggest to overcome this obstruction by studying families of Vassiliev invariants which, roughly, are associated to finite coverings of $M$. In the case $…
We construct a new class of finite-dimensional C^*-quantum groupoids at roots of unity q=e^{iπ/\ell}, with limit the discrete dual of the classical SU(N) for large orders. The representation category of our groupoid turns out to be tensor equivalent to the well known quotient C^*-category of the category of tilting modules of the non-semisimple quantum group U_q({\mathfrak sl}_N) of Drinfeld, Jimbo and Lusztig. As an algebra, the C^*-groupoid is a quotient of U_q({\mathfrak sl}_N). As a coalgebra, it naturally reflects the categorical quotient construction. In particular, it is not coassociati…
Deformations of mixed associators in module categories
Institut de Mathématiques de Toulouse, Université Paul Sabatier（图卢兹数学研究所，保罗萨巴蒂尔大学）；Institut Denis Poisson, CNRS, Université de Tours, Université d’Orléans（丹尼斯普瓦松研究所，法国国家科学研究中心，图尔大学，奥尔良大学）；Fachbereich Mathematik, Universität Hamburg（汉堡大学数学系）
