Related papers: Quantum Teichmuller theory and representations of …
When a quantum hyperboloid is realized, as a three - parameter algebra $\ahqc$, in the usual manner, the following problem arises: what is a ``representation theory'' of this algebra? We construct the series of all spin representations of…
This paper gives a new, simplified presentation of the classical pure braid group. The generators are given by the squares of the longest elements over connected subgraphs, and we prove that the only relations are either commutators or…
The topological model for quantum computation is an inherently fault-tolerant model built on anyons in topological phases of matter. A key role is played by the braid group, and in this survey we focus on a selection of ways that the…
We study representations of the loop braid group $LB_n$ from the perspective of extending representations of the braid group $B_n$. We also pursue a generalization of the braid/Hecke/Temperlely-Lieb paradigm---uniform finite dimensional…
We unify and generalize several approaches to constructing braid group representations from finite groups, using iterated twisted tensor products. Our results hint at a relationship between the braidings on the $G$-gaugings of a pointed…
We introduce the notion of a braided Lie algebra consisting of a finite-dimensional vector space $\CL$ equipped with a bracket $[\ ,\ ]:\CL\tens\CL\to \CL$ and a Yang-Baxter operator $\Psi:\CL\tens\CL\to \CL\tens\CL$ obeying some axioms. We…
In a recent paper the quantum 2-sphere $S^2_q$ was described as a quantum complex manifold. Here we consider several copies of $S^2_q$ and derive their braiding commutation relations. The braiding is extended to the differential and to the…
We construct a braided analogue of the quantum permutation group and show that it is the universal braided compact quantum group acting on a finite space in the category of $\mathbb{Z}/N\mathbb{Z}$-$\textrm{C}^*$-algebras with a twisted…
We study and give examples of braided groupoids, and, a fortiori, non-degenerate solutions of the quiver-theoretical braid equation.
We clarify some aspects of quantum group gauge theory and its recent generalisations (by T. Brzezinski and the author) to braided group gauge theory and coalgebra gauge theory. We outline the diagrammatic version of the braided case. We…
We define three families of quivers in which the braid relations of the symmetric group $S_n$ are realized by mutations and automorphisms. A sequence of eight braid moves on a reduced word for the longest element of $S_4$ yields three…
We describe all fusion subcategories of the representation category of a twisted quantum double of a finite group. In view of the fact that every group-theoretical braided fusion category can be embedded into a representation category of a…
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…
Representations of small quantum groups $u_q({\mathfrak{g}})$ at a root of unity and their extensions provide interesting tensor categories, that appear in different areas of algebra and mathematical physics. There is an ansatz by Lusztig…
We propose a family of new representations of the braid groups on surfaces that extend linear representations of the braid groups on a disc such as the Burau representation and the Lawrence-Krammer-Bigelow representation.
This is a systematic introduction for physicists to the theory of algebras and groups with braid statistics, as developed over the last three years by the author. There are braided lines, braided planes, braided matrices and braided groups…
We present an explicit form of braided symmetries of the quantum spheres, by introducing a braided quantum Hopf algebra $\cU_{q, \phi}$ and demonstrating that they are braided Hopf modules over this braided Hopf algebra. To obtain this…
We produce an explicit description of the K-theory and K-homology of the pure braid group on $n$ strands. We describe the Baum--Connes correspondence between the generators of the left- and right-hand sides for $n=4$. Using functoriality of…
In this paper, we study quasi-homomorphisms of quantum cluster algebras, which are quantum analogy of quasi-homomorphisms of cluster algebras introduced by Fraser. For a quantum Grassmannian cluster algebra $\mathbb{C}_q[{\rm Gr}(k,n)]$, we…
The Standard Model of particle physics provides very accurate predictions of phenomena occurring at the sub-atomic level, but the reason for the choice of symmetry group and the large number of particles considered elementary, is still…