Related papers: On Drinfel'd associators
Drinfel'd used associators to construct families of universal representations of braid groups. We consider semi-associators (i.e., we drop the pentagonal axiom and impose a normalization in degree one). We show that the process may be…
This work presents an approach towards the representation theory of the braid groups $B_n$. We focus on finite-dimensional representations over the field of Laurent series which can be obtained from representations of infinitesimal braids,…
We present a formalism within which the relationship (discovered by Drinfel'd) between associators (for quasi-triangular quasi-Hopf algebras) and (a variant of) the Grothendieck-Teichmuller group becomes simple and natural, leading to a…
We identify Drinfeld's formal group on the prismatization of $\mathrm{Spf}\,\mathbb{Z}_p$ with a formal group arising from homotopy theory, given locally by the Quillen formal group of a decompleted variant of topological periodic cyclic…
We describe the general non-associative version of Lie theory that relates unital formal multiplications (formal loops), Sabinin algebras and non-associative bialgebras. Starting with a formal multiplication we construct a non-associative…
These notes refer to a minicourse I gave at the occasion of the conference meeting ``Applications of Noncommutative Geometry to Gauge Theories, Field Theories, and Quantum Space-Time'' to be held from 7 April to 11 April 2025 at the Centre…
Drinfeld orbifold algebras are a type of deformation of skew group algebras generalizing graded Hecke algebras of interest in representation theory, algebraic combinatorics, and noncommutative geometry. In this article, we classify all…
We develop an extension of the usual theory of formal group laws where the base ring is not required to be commutative and where the formal variables need neither be central nor have to commute with each other. We show that this is the…
In this note we give an introduction to Drinfel'd's associator coming from the Knizhnik-Zamolodchikov connections and a self-contained proof of the hexagon and pentagon equations by means of minimal amounts of analysis or differential…
We solve the regularized Knizhnik-Zamolodchikov equation and find an explicit expression for the Drinfeld associator. We restrict to the case of the fundamental representation of $gl(N)$. Several tests of the results are presented. It can…
The set of formal power series with coefficients in an associative but noncommutative algebra becomes a loop with the substitution product. We initiate the study of this loop by describing certain Lie and Sabinin algebras related to it.…
Chiral conformal blocks in a rational conformal field theory are a far going extension of Gauss hypergeometric functions. The associated monodromy representations of Artin's braid group capture the essence of the modern view on the subject,…
We compute explicitly the monodromy representations of "cyclotomic" analogs of the Knizhnik--Zamolodchikov differential system. These are representations of the type B braid group $B_n^1$. We show how the representations of the braid group…
We establish an equivalence between categories of 'formally nilpotent' Lie algebras and exponential groups in characteristic zero. It extends the equivalences of Mal'cev, Lazard, Quillen and Warfield, and applies to groups under composition…
We define a new homotopy algebraic structure, that we call a braided $L_\infty$-algebra, and use it to systematically construct a new class of noncommutative field theories, that we call braided field theories. Braided field theories have…
In this work we study nonuniform exponential dichotomies and existence of pullback and forward attractors for evolution processes associated to nonautonomous differential equations. We define a new concept of nonuniform exponential…
For the solutions $\Phi(z)$ of functional equations $\Phi(z)=P(z)+\Phi(Q(z))$, we derive a complete asymptotic of power series coefficients. As an application, we improve significantly an asymptotic of the number of $2,3$-trees with $n$…
We study conjugacy of formal derivations on fields of generalised power series in characteristic 0. Casting the problem of Poincar\'e resonance in terms of asymptotic differential algebra, we give conditions for conjugacy of parabolic flat…
These are lecture notes prepared for a minicourse given at the Cimpa Research School "Algebraic and geometric aspects of representation theory", held in Curitiba, Brazil in March 2013. The purpose of the course is to provide an introduction…
Drinfeld twists, and the twists of Giaquinto and Zhang, allow for algebras and their modules to be deformed by a cocycle. We prove general results about cocycle twists of algebra factorisations and induced representations and apply them to…