Cyclotomic associators and finite type invariants for tangles in the solid torus

  title={Cyclotomic associators and finite type invariants for tangles in the solid torus},
  author={Adrien Brochier},
  journal={arXiv: Quantum Algebra},
  • A. Brochier
  • Published 3 September 2012
  • Mathematics
  • arXiv: Quantum Algebra
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… 

Minimal nondegenerate extensions

We prove that every slightly degenerate braided fusion category admits a minimal nondegenerate extension. As a corollary, every pseudounitary super modular tensor category admits a minimal modular

On the Classification of Topological Orders

We axiomatize the extended operators in topological orders (possibly gravitationally anomalous, possibly with degenerate ground states) in terms of monoidal Karoubi-complete $n$-categories which are

Contributions to the theory of KZB associators

In this thesis, following the work initiated by V. Drinfeld and pursued by B. Enriquez, then by the latter together with D. Calaque and P. Etingof, we study the universal twisted elliptic

A topological origin of quantum symmetric pairs

It is well known that braided monoidal categories are the categorical algebras of the little two-dimensional disks operad. We introduce involutive little disks operads, which are $${\mathbb

Quantum character varieties and braided module categories

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants $$\int _S{\mathcal {A}}$$∫SA of a surface S, determined by the

Finite symmetries of quantum character stacks

For a finite group D, we study categorical factorisation homology on oriented surfaces equipped with principal D-bundles, which ‘integrates’ a (linear) balanced braided category A with D-action over

Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik-Zamolodchikov equations and Letzter-Kolb coideals

We establish an equivalence between two approaches to quantization of irreducible symmetric spaces of compact type within the framework of quasi-coactions, one based on the Enriquez-Etingof

Braided Picard groups and graded extensions of braided tensor categories

We classify various types of graded extensions of a finite braided tensor category $\cal B$ in terms of its $2$-categorical Picard groups. In particular, we prove that braided extensions of $\cal B$

A categorical approach to dynamical quantum groups

Abstract We present a categorical point of view on dynamical quantum groups in terms of categories of Harish-Chandra bimodules. We prove Tannaka duality theorems for forgetful functors into the

Braided module categories via quantum symmetric pairs

  • S. Kolb
  • Mathematics
    Proceedings of the London Mathematical Society
  • 2019
Let g be a finite‐dimensional complex semisimple Lie algebra. The finite‐dimensional representations of the quantized enveloping algebra Uq(g) form a braided monoidal category Oint . We show that the



Quasi-reflection algebras and cyclotomic associators

Abstract.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 \rightarrow

Universal KZB Equations: The Elliptic Case

We define a universal version of the Knizhnik–Zamolodchikov–Bernard (KZB) connection in genus 1. This is a flat connection over a principal bundle on the moduli space of elliptic curves with marked


Following Drinfel’d, Kontsevich and Piunikhin, we study the iterated integral expression for the holonomy of the formal Knizhnik-Zamolodchikov connection, finding that by introducing non-associative

Chord diagram invariants of tangles and graphs

The notion of a chord diagram emerged from Vassiliev's work Vas90], Vas92] (see also Gusarov Gus91], Gus94] and Bar-Natan BN91], BN95]). Slightly later, Kontsevich Kon93] deened an invariant of

Knot theory related to generalized and cyclotomic Hecke algebras of type ℬ

In [12] is established that knot isotopy in a 3-manifold may be interpreted in terms of Markov braid equivalence and, also, that the braids related to the 3-manifold form algebraic structures.


We study Vassiliev invariants of links in a 3-manifold M by using chord diagrams labeled by elements of the fundamental group of M. We construct universal Vassiliev invariants of links in M, where

Claspers and finite type invariants of links.

We introduce the concept of \claspers," which are surfaces in 3{manifolds with some additional structure on which surgery operations can be performed. Using claspers we dene for each positive integer

A Kohno–Drinfeld Theorem for the Monodromy of Cyclotomic KZ Connections

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

On the Vassiliev knot invariants


0. Introduction. Quantum invariants of framed links whose components are colored by modules of a simple Lie algebra g are Laurent polynomials inv1/D (with integer coefficients), where v is the