Standard objects in 2-braid groups (original) (raw)

On Braided Linear Gr-categories

We provide explicit and unified formulae for the normalized 3-cocycles on arbitrary finite abelian groups. As an application, we compute all the braided monoidal structures on linear Gr-categories.

On some groups related to the Braid Groups of type A

Annals of the University of Craiova - Mathematics and Computer Science Series, 2010

We prove that a family of groups R(n) forms the algebraic structure of an operad and that they admit a presentation similar to that of the Braid groups of type A. This result provides a new proof that the Braid Groups form an operad, a topic emphasized in ~\cite{16}~\cite{ulrike}. These groups proved to be useful in several problems which belong to different areas of Mathematics. Representations of R(n) came from a system of mixed Yang-Baxter type equations. We define the Hopf equation in braided monoidal categories and we prove that representations for our groups came from any braided Hopf algebra with invertible antipode. Using this result, we prove that there is a morphism from R(n) to the mapping class group Gamman,1\Gamma_{n,1}Gamman,1, using some results from 3-dimensional topology.

Abstract commensurators of braid groups

Journal of Algebra, 2006

Let B n be the braid group on n ≥ 4 strands. We show that the abstract commensurator of B n is isomorphic to Mod(S) ⋉ (Q × ⋉ Q ∞ ), where Mod(S) is the extended mapping class group of the sphere with n + 1 punctures.

The pure braid groups and their relatives

In this talk, I discuss the resonance varieties, the lower central series ranks, the Chen ranks, and the formality properties of several families of braid-like groups: the pure braid groups P_n, the welded pure braid groups wP_n, the virtual pure braid groups vP_n, as well as their `upper' variants, wP_n^+ and vP_n^+. I will also discuss several natural homomorphisms between these groups, and various ways to distinguish among the pure braid groups and their relatives. This is joint work with He Wang.

The BNS-invariant for the pure braid groups

Groups, Geometry, and Dynamics, 2015

In Bieri, Neumann and Strebel introduced a geometric invariant for discrete groups. In this article we compute and explicitly describe the BNS-invariant for the pure braid groups.

Some remarks on braided group reconstruction and braided doubles

1998

The cross coproduct braided group Aut(C)◮ <B is obtained by Tannaka-Krein reconstruction from C B → C for a braided group B in braided category C. We apply this construction to obtain partial solutions to two problems in braided group theory, namely the tensor problem and the braided double. We obtain Aut(C)◮ <Aut(C) ∼ =Aut(C)>⊳Aut(C) and higher braided group 'spin chains'. The example of the braided group B(R)>⊳B(R)>⊳ • • • >⊳B(R) is described explicitly by R-matrix relations. We also obtain Aut(C)◮ <Aut(C) * as a dual quasitriangular 'codouble' braided group by reconstruction from the dual category C • → C. General braided double crossproducts B⊲⊳C are also considered.

2 Dual Garside Structure and Reducibility of Braids

2016

Benardete, Gutierrez and Nitecki showed an important result which relates the geometrical properties of a braid, as a homeomorphism of the punctured disk, to its algebraic Garside-theoretical properties. Namely, they showed that if a braid sends a standard curve to another standard curve, then the image of this curve under the action of each factor of the left normal form of the braid (with the classical Garside structure) is also standard. We provide a new simple, geometric proof of the result by Benardete-Gutierrez-Nitecki, which can be easily adapted to the case of the dual Garside structure of braid groups, with the appropriate definition of standard curves in the dual setting. This yields a new algorithm for determining the Nielsen-Thurston type of braids.

On the Linear Representation of Braid Groups

Transactions of the American Mathematical Society, 1989

We give a new derivative of the Burau and Gassner representations of the braid and pure braid groups. Various applications are explored.

Some monodromy representations of generalized braid groups

Communications in Mathematical Physics, 1994

A flat connection on the trivial bundle over the complement in C" of the complexification of the system of the reflecting hyperplanes of the B n , D n Coxeter groups is built from a simple Lie algebra and its representation. The corresponding monodromy representations of the generalized braid groups XB n , XD n are computed in the simplest case.

On monoidal functors between (braided) Gr-categories

Arxiv preprint arXiv:1105.5186, 2011

Abstract: In this paper, we state and prove precise theorems on the classification of the category of (braided) categorical groups and their (braided) monoidal functors, and some applications obtained from the basic studies on monoidal functors between categorical ...