Related papers: Blocked-braid Groups

We initiate a general approach to the relative braid group symmetries on (universal) $\imath$quantum groups, arising from quantum symmetric pairs of arbitrary finite types, and their modules. Our approach is built on new intertwining…

We prove a conjecture due to Makanin: if a and b are elements of the Artin braid group B_n such that a^k=b^k for some nonzero integer k, then a and b are conjugate. The proof involves the Nielsen-Thurston classification of braids.

We consider the universal family $E_n^d$ of superelliptic curves: each curve $\Sigma_n^d$ in the family is a $d$-fold covering of the unit disk, totally ramified over a set $P$ of $n$ distinct points; $\Sigma_n^d\hookrightarrow E_n^d\to…

We study representations of the braid groups from braiding gapped boundaries of Dijkgraaf-Witten theories and their twisted generalizations, which are (twisted) quantum doubled topological orders in two spatial dimensions. We show that the…

We consider quantum group theory on the Hilbert space level. We find all unitary representations of three braided quantum groups related to the quantum ``ax+b'' group. First we introduce an auxiliary braided quantum group, which is…

This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise super-algebras and super-Hopf algebras, aswell as colour-Lie algebras. Basic facts about braided categories C…

We study the Brauer groups of regular conic bundles over elliptic curves defined over a number field $k$. We explicitly compute the Brauer group of the conic bundle when the singular fibres lie above $k$-points that are divisible by $2$ in…

We compute the braided groups and braided matrices $B(R)$ for the solution $R$ of the Yang-Baxter equation associated to the quantum Heisenberg group. We also show that a particular extension of the quantum Heisenberg group is dual to the…

We interpret a recent formula for counting orbits of $GL(d,F_q)$ in terms of counting fixed points as addition in the affine braided line. The theory of such braided groups (or Hopf algebras in braided categories) allows us to obtain the…

Since the braid group was discovered by E. Artin, the question of its conjugacy problem has been solved by Garside and Birman, Ko and Lee. However, the solutions given thus far are difficult to compute with a computer, since the number of…

We give a complete classification of homomorphisms from the commutator subgroup of the braid group on $n$ strands to the braid group on $n$ strands when $n$ is at least 7. In particular, we show that each nontrivial homomorphism extends to…

Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. They are a class of algebras with triangular decomposition, arising from a deformation problem, the…

Suppose $X$ is a torsor under an abelian variety $A$ over a number field. We show that any adelic point of $X$ that is orthogonal to the algebraic Brauer group of $X$ is orthogonal to the whole Brauer group of $X$. We also show that if…

By evaluating the Burau representation at t=-1, we obtain a symplectic representation of the braid group. We define the congruence subgroups of the braid group to be the preimages of the principal congruence subgroups of the symplectic…

We define an action of the braid group of a simple Lie algebra on the space of imaginary roots in the corresponding quantum affine algebra. We then use this action to determine an explicit condition for a tensor product of arbitrary…

Let $(\mathrm{U},\mathrm{U}^\imath)$ be the quantum symmetric pair of arbitrary finite type and $G^*$ be the associated dual Poisson-Lie group. Generalizing the work of De Concini and Procesi, the first author introduced an integral form…

Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic from the braid group to the mapping class group. We prove here that this map is trivial in stable homology with…

We study the representations of the commutator subgroup K_{n} of the braid group B_{n} into a finite group . This is done through a symbolic dynamical system. Some experimental results enable us to compute the number of subgroups of K_{n}…

A near-group category is an additively semisimple category with a product such that all but one of the simple objects is invertible. We classify braided structures on near-group categories, and give explicit numerical formulas for their…

We introduce and develop a language of semigroups over the braid groups for a study of braid monodromy factorizations (bmf's) of plane algebraic curves and other related objects. As an application we give a new proof of Orevkov's theorem on…