Related papers: Braids as a representation space of SU(5)
We give an overview of a new kind symmetry in physics which exists between observables and states and which is made possible by the language of Hopf algebras and quantum geometry. It has been proposed by the author as a feature of Planck…
We first motivate the study of a certain quotient of the loop braid category, both for the mathematics underpinning recent approaches to topological quantum computation; and as a key example in non-semisimple higher representation theory.…
We introduce a representation theory of q-Lie algebras defined earlier in \cite{DG1}, \cite{DG2}, formulated in terms of braided modules. We also discuss other ways to define Lie algebra-like objects related to quantum groups, in…
Quantum contextuality, a fundamental feature distinguishing quantum theory from classical models, is investigated via algebraic and topological structures inherent in modular tensor categories. This work rigorously demonstrates that braid…
Division algebras have demonstrated their utility in studying non-associative algebras and their connection to the Standard Model through complex Clifford algebras. This article focuses on exploring the connection between these complex…
We study homological representations of mapping class groups, including the braid groups. These arise from the twisted homology of certain configuration spaces, and come in many different flavours. Our goal is to give a unified general…
Recent work on neutrino oscillations suggests that the three generations of fermions in the standard model are related by representations of the finite group A(4), the group of symmetries of the tetrahedron. Motivated by this, we explore…
Postulating that spacetime is discrete, we assume that physical space is described by a 3-dimensional cubic lattice.The corresponding symmetry group of rotations has order 24 and motivates the introduction of a cubic shaped graph with 27…
Braid groups are an important and flexible tool used in several areas of science, such as Knot Theory (Alexander's theorem), Mathematical Physics (Yang-Baxter's equation) and Algebraic Geometry (monodromy invariants). In this note we will…
Loop braid groups characterize the exchange of extended objects, namely loops, in three dimensional space generalizing the notion of braid groups that describe the exchange of point particles in two dimensional space. Their interest in…
We use the theory of regular objects in tensor categories to clarify the passage between braided multiplicative unitaries and multiplicative unitaries with projection. The braided multiplicative unitary and its semidirect product…
Drinfel'd used associators to construct families of universal representations of braid groups. We consider semi-associators (i.e., we drop the pentagonal axiom and impose a normalization in degree one). We show that the process may be…
Path algebras are a convenient way of describing decompositions of tensor powers of an object in a tensor category. If the category is braided, one obtains representations of the braid groups $B_n$ for all $n\in \N$. We say that such…
We study a wide range of homologically-defined representations of surface braid groups and of mapping class groups of surfaces, including the Lawrence-Bigelow representations of the classical braid groups. These representations naturally…
String and particle braiding statistics are examined in a class of topological orders described by discrete gauge theories with a gauge group $G$ and a 4-cocycle twist $\omega_4$ of $G$'s cohomology group…
A braided generalization of the concept of Hopf algebra (quantum group) is presented. The generalization overcomes an inherent geometrical inhomogeneity of quantum groups, in the sense of allowing completely pointless objects. All…
Topological quantum computation employs two-dimensional quasiparticles called anyons. The generally accepted mathematical basis for the theory of anyons is the framework of modular tensor categories. That framework involves a substantial…
We study a novel type of braid groups on a closed orientable surface $\Sigma$. These are fundamental groups of certain manifolds that are hybrids between symmetric products and configuration spaces of points on $\Sigma$; a class of examples…
Linearity allows several versions of reality to simultaneously exist in the state vector. But it implies that there is no interaction between versions, and that there will never be perception of more than one version. It also implies, in…
We consider the braid group representation which describes the non-abelian braiding statistics of the spin 1/2 particle world lines of an SU(2)$_4$ Chern-Simons theory. Up to an abelian phase, this is the same as the non-Abelian statistics…