scholarly journals A CATEGORICAL MODEL FOR THE VIRTUAL BRAID GROUP

2012 ◽  
Vol 21 (13) ◽  
pp. 1240008 ◽  
Author(s):  
LOUIS H. KAUFFMAN ◽  
SOFIA LAMBROPOULOU
Keyword(s):  
Braid Group ◽  
Hopf Algebras ◽  
Braid Groups ◽  
Point Of View ◽  
Baxter Equation ◽  
Quantum Invariants ◽  
Natural Group ◽  
Virtual Links ◽  
New Interpretation ◽  
Pure Virtual Braid Groups

This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this approach is to take pure virtual braids as primary. The generators of the pure virtual braid group are abstract solutions to the algebraic Yang–Baxter equation. This point of view illuminates representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter equation. In this categorical framework, the virtual braid group is a natural group associated with the structure of algebraic braiding. We then point out how the category SC is related to categories associated with quantum algebras and Hopf algebras and with quantum invariants of virtual links.

On marked braid groups

2015 ◽  
Vol 24 (13) ◽  
pp. 1541005 ◽  
Author(s):  
Denis A. Fedoseev ◽  
Vassily O. Manturov ◽  
Zhiyun Cheng
Keyword(s):  
Braid Group ◽  
Knot Theory ◽  
Braid Groups ◽  
Arbitrary Group ◽  
Natural Group ◽  
Additional Information ◽  
Reidemeister Moves ◽  
Braid Theory ◽  
Important Notion

In this paper, we introduce [Formula: see text]-braids and, more generally, [Formula: see text]-braids for an arbitrary group [Formula: see text]. They form a natural group-theoretic counterpart of [Formula: see text]-knots, see [V. O. Manturov; Reidemeister moves and groups, preprint (2014), arXiv:1412.8691]. The underlying idea used in the construction of these objects — decoration of crossings with some additional information — generalizes an important notion of parity introduced by the second author (see [V. O. Manturov, Parity in knot theory, Sb. Math. 201(5) (2010) 693–733]) to different combinatorically geometric theories, such as knot theory, braid theory and others. These objects act as natural enhancements of classical (Artin) braid groups. The notion of dotted braid group is introduced: classical (Artin) braid groups live inside dotted braid groups as those elements having presentation with no dots on the strands. The paper is concluded by a list of unsolved problems.

scholarly journals QUASITRIANGULAR HOPF ALGEBRAS, BRAID GROUPS AND QUANTUM ENTANGLEMENT

2013 ◽  
Vol 11 (07) ◽  
pp. 1350065 ◽  
Author(s):  
ERIC PINTO ◽  
MARCO A. S. TRINDADE ◽  
J. D. M. VIANNA
Keyword(s):  
Quantum Logic ◽  
Quantum Entanglement ◽  
Braid Group ◽  
Hopf Algebras ◽  
Logic Gates ◽  
Braid Groups ◽  
Group Algebras ◽  
Algebraic Structures ◽  
Quantum Logic Gates ◽  
Cyclic Groups

The aim of the paper is to provide a method to obtain representations of the braid group through a set of quasitriangular Hopf algebras. In particular, these algebras may be derived from group algebras of cyclic groups with additional algebraic structures. In this context, by using the flip operator, it is possible to construct R-matrices that can be regarded as quantum logic gates capable of preserving quantum entanglement.

scholarly journals Geometric Presentations of Braid Groups for Particles on a Graph

2021 ◽  
Author(s):  
Byung Hee An ◽  
Tomasz Maciazek
Keyword(s):  
Braid Group ◽  
Planar Graphs ◽  
Braid Groups ◽  
Quantum Statistics ◽  
Simple Cycle ◽  
Full Description ◽  
Baxter Equation ◽  
Connected Graphs ◽  
Star Graphs ◽  
Graph Theoretic

AbstractWe study geometric presentations of braid groups for particles that are constrained to move on a graph, i.e. a network consisting of nodes and edges. Our proposed set of generators consists of exchanges of pairs of particles on junctions of the graph and of certain circular moves where one particle travels around a simple cycle of the graph. We point out that so defined generators often do not satisfy the braiding relation known from 2D physics. We accomplish a full description of relations between the generators for star graphs where we derive certain quasi-braiding relations. We also describe how graph braid groups depend on the (graph-theoretic) connectivity of the graph. This is done in terms of quotients of graph braid groups where one-particle moves are put to identity. In particular, we show that for 3-connected planar graphs such a quotient reconstructs the well-known planar braid group. For 2-connected graphs this approach leads to generalisations of the Yang–Baxter equation. Our results are of particular relevance for the study of non-abelian anyons on networks showing new possibilities for non-abelian quantum statistics on graphs.

scholarly journals EXTRASPECIAL 2-GROUPS AND IMAGES OF BRAID GROUP REPRESENTATIONS

2006 ◽  
Vol 15 (04) ◽  
pp. 413-427 ◽  
Author(s):  
JENNIFER M. FRANKO ◽  
ERIC C. ROWELL ◽  
ZHENGHAN WANG
Keyword(s):  
Finite Groups ◽  
Braid Group ◽  
Braid Groups ◽  
Baxter Equation ◽  
Group Representations ◽  
Specific Solution ◽  
Link Invariants

We investigate a family of (reducible) representations of the braid groups [Formula: see text] corresponding to a specific solution to the Yang–Baxter equation. The images of [Formula: see text] under these representations are finite groups, and we identify them precisely as extensions of extra-special 2-groups. The decompositions of the representations into their irreducible constituents are determined, which allows us to relate them to the well-known Jones representations of [Formula: see text] factoring over Temperley–Lieb algebras and the corresponding link invariants.

STRANGE STATISTICS, BRAID GROUP REPRESENTATIONS AND MULTIPOINT FUNCTIONS IN THE N-COMPONENT MODEL

1989 ◽  
Vol 04 (09) ◽  
pp. 2333-2370 ◽  
Author(s):  
H. C. LEE ◽  
M. L. GE ◽  
M. COUTURE ◽  
Y. S. WU
Keyword(s):  
Braid Group ◽  
Point Of View ◽  
Component Model ◽  
Baxter Equation ◽  
Many Body ◽  
Group Representations ◽  
Conformal Field ◽  
Many Body Systems ◽  
Group B ◽  
Bose Einstein

The statistics of fields in low dimensions is studied from the point of view of the braid group Bn of n strings. Explicit representations MR for the N-component model, N=2 to 5, are derived by solving the Yang-Baxter-like braid group relations for the statistical matrix R, which describes the transformation of the bilinear product of two N-component fields under the transposition of coordinates. When R2≠1 the statistics is neither Bose-Einstein nor Fermi-Dirac; it is strange. It is shown that for each N, the N+1 parameter family of solutions obtained is the most general one under a given set of constraints including “charge” conservation. Extended Nth order (N>2) Alexander-Conway relations for link polynomials are derived. They depend nonhomogeneously only on one of the N+1 parameters. The N=3 and 4 ones agree with those previously derived by Akutsu et al. Flat connections ω defining integrable systems of the N-component model are derived from the representations. The monodromy of the solution of such a system also carries a representation Mω of Pn⊂Bn. For N=2, Mω=MR, but the equality may not hold in general. The connections also lead directly to solutions of the classical Yang-Baxter equation. A generalization of Kohno’s monodromy representation of Bn associated with the algebra sl(N,C) is given. Applications of the braid group representations to statistical models, conformal field theory and many-body systems of extended objects are briefly discussed.

scholarly journals Finiteness for mapping class group representations from twisted Dijkgraaf–Witten theory

2018 ◽  
Vol 27 (06) ◽  
pp. 1850043 ◽  
Author(s):  
Paul P. Gustafson
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Braid Groups ◽  
Fusion Category ◽  
Mapping Class ◽  
Group Representations ◽  
Colored Graphs ◽  
Finite Image ◽  
Spanning Set

We show that any twisted Dijkgraaf–Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof et al. showing that the braid group images are finite [P. Etingof, E. C. Rowell and S. Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math. 234 (2008)(1) 33–42]. In particular, our result answers their question regarding finiteness of images of arbitrary mapping class group representations in the affirmative. Our approach is to translate the problem into manipulation of colored graphs embedded in the given surface. To do this translation, we use the fact that any twisted Dijkgraaf–Witten representation associated to a finite group [Formula: see text] and 3-cocycle [Formula: see text] is isomorphic to a Turaev–Viro–Barrett–Westbury (TVBW) representation associated to the spherical fusion category [Formula: see text] of twisted [Formula: see text]-graded vector spaces. The representation space for this TVBW representation is canonically isomorphic to a vector space of [Formula: see text]-colored graphs embedded in the surface [A. Kirillov, String-net model of Turaev-Viro invariants, Preprint (2011), arXiv:1106.6033 ]. By analyzing the action of the Birman generators [J. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213–242] on a finite spanning set of colored graphs, we find that the mapping class group acts by permutations on a slightly larger finite spanning set. This implies that the representation has finite image.

AUTOMORPHISMS OF BRAID GROUPS ON S2, T2, P2 AND THE KLEIN BOTTLE K

2008 ◽  
Vol 17 (01) ◽  
pp. 47-53 ◽  
Author(s):  
PING ZHANG
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Klein Bottle ◽  
Closed Surface ◽  
Group Extension ◽  
Braid Groups ◽  
Mapping Class ◽  
Group B ◽  
Central Automorphisms

It is shown that for the braid group Bn(M) on a closed surface M of nonnegative Euler characteristic, Out (Bn(M)) is isomorphic to a group extension of the group of central automorphisms of Bn(M) by the extended mapping class group of M, with an explicit and complete description of Aut (Bn(S2)), Aut (Bn(P2)), Out (Bn(S2)) and Out (Bn(P2)).

Braid lift representations of Artin's Braid Group

2000 ◽  
Vol 09 (08) ◽  
pp. 1005-1009
Author(s):  
Reinhard Häring-Oldenburg
Keyword(s):  
Braid Group ◽  
Braid Groups ◽  
Finite Dimensional ◽  
Braid Theory

We recast the braid-lift representation of Contantinescu, Lüdde and Toppan in the language of B-type braid theory. Composing with finite dimensional representations of these braid groups we obtain various sequences of finite dimensional multi-parameter representations.

scholarly journals Constructing Hopf braces

2019 ◽  
Vol 30 (02) ◽  
pp. 1850089 ◽  
Author(s):  
A. L. Agore
Keyword(s):  
Hopf Algebras ◽  
Baxter Equation ◽  
Matched Pairs

We investigate Hopf braces, a concept recently introduced by Angiono, Galindo and Vendramin [I. Angiono, C. Galindo and L. Vendramin, Hopf braces and Yang–Baxter operators, Proc. Amer. Math. Soc. 145 (2017) 1981–1995] in connection to the quantum Yang–Baxter equation. More precisely, we propose two methods for constructing Hopf braces. The first one uses matched pairs of Hopf algebras while the second one relies on category-theoretic tools.

scholarly journals Negative curvature in graph braid groups

2020 ◽  
pp. 1-36
Author(s):  
Anthony Genevois
Keyword(s):  
Braid Group ◽  
Negative Curvature ◽  
Braid Groups ◽  
Gromov Hyperbolic ◽  
Relatively Hyperbolic ◽  
Geometric Study

In this paper, we initiate a geometric study of graph braid groups. More precisely, by applying the formalism of special colorings introduced in a previous paper, we determine precisely when a graph braid group is Gromov-hyperbolic, toral relatively hyperbolic and acylindrically hyperbolic.

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