覆盖在$G_2$张量空间上的辫群

Classification of charge-conserving loop braid representations

Here a loop braid representation is a monoidal functor $\mathsf{F}$ from the loop braid category $\mathsf{L}$ to a suitable target category, and is $N$-charge-conserving if that target is the category $\mathsf{Match}^N$ of charge-conserving matrices (specifically $\mathsf{Match}^N$ is the same rank-$N$ charge-conserving monoidal subcategory of the monoidal category $\mathsf{Mat}$ used to classify braid representations in arXiv:2112.04533) with $\mathsf{F}$ strict, and surjective on $\mathbb{N}$, the object monoid. We classify and construct all such representations. In particular we prove that …

矢量编织

Vector Braids

本文定义了一类新的群族，它推广了$\C $上的{\it经典辫群。我们用$\{B_n^m\}_{n \ge m+1}$表示这个族，其中$n,m \in \N$。家族$\{ B_n^1 \}_{n \in \N}$是$n$弦上的经典辫子群的集合。群$B_n^m$是$\C^m$中$n$无序点的运动集合，使得在运动期间的任何时刻，点的每个$m+1$跨越作为仿射空间的整个$\C^m$。有一个从$B_n^m$到$n$字母上对称群的映射。我们让$P_n^m$表示这个映射的核。在本文中，我们主要有兴趣了解$P_n^2$。给出了一个满射映射到$P_n^2$上的群$PL_n$.我们还证明了满射$PL_n \to P_n^2$在第一和第二积分同调上诱导同构，并猜想它是同构。然后，我们发现一个无穷小的介绍组$P_n^2$。最后，我们还考虑了点位于$¶^m$而不是$\C^m$中的类比群。这些群推广了球面上的经典辫群。

辫子霍普夫代数上的弱投影

Weak Projections onto a Braided Hopf Algebra

我们证明了，在一些温和的条件下，一个交换和coabel辫状幺半范畴中的双代数有一个弱投影到一个形式光滑（作为余代数）的子双代数上，该子双代数具有对极，见定理1.12。本文第二部分证明了具有弱投影的双代数是向量积双代数，见定理2.12。当双代数$A$是余交换的且与弱投影相关的某个余圈是平凡的时，我们证明了$A$是双向量积，或Madjid术语中的双积.最后一个结果是基于双向量积的一个泛性质，根据定理2.15，它适用于辫状幺半范畴范畴。我们还调查的情况下，正确的行动相关联的匹配对是平凡的。

The Soergel category and the redotted Webster algebra

We describe a collection of graded rings which surject onto Webster rings for sl(2) and which should be related to certain categories of singular Soergel bimodules. In the first non-trivial case, we construct a categorical braid group action which categorifies the Burau representation.

Diagram complexes, formality, and configuration space integrals for braids

We use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chen's iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a CDGA suitable for integration. We show that this integration is compatible with Bott-Taubes configuration space integrals for long l…

Congruence subgroups from representations of the three-strand braid group

Ng and Schauenburg proved that the kernel of a $(2+1)$-dimensional topological quantum field theory representation of $\mathrm{SL}(2, \mathbb{Z})$ is a congruence subgroup. Motivated by their result, we explore when the kernel of an irreducible representation of the braid group $B_3$ with finite image enjoys a congruence subgroup property. In particular, we show that in dimensions two and three, when the projective order of the image of the braid generator $σ_1$ is between 2 and 5 the kernel projects onto a congruence subgroup of $\mathrm{PSL}(2,\mathbb{Z})$ and compute its level. However, we …