Related papers: Method of constructing braid group representation …
Entangled states, such as the Bell and GHZ states, are generated from separable states using matrices known to satisfy the Yang-Baxter equation and its generalization. This remarkable fact hints at the possibility of using braiding…
We study the rational solution of the Yang-Baxter equation with the supersymmetry algebra sl(2|1). The R-matrix acting in the tensor product of two arbitrary representations of the supersymmetry algebra can be represented as the product of…
A coloured braid group representation (CBGR) is constructed with the help of some modified universal ${\cal R}$-matrix, associated to $U_q(gl(2))$ quantised algebra. Explicit realisation of Faddeev-Reshetikhin-Takhtajan (FRT) algebra is…
This work presents an approach towards the representation theory of the braid groups $B_n$. We focus on finite-dimensional representations over the field of Laurent series which can be obtained from representations of infinitesimal braids,…
Complete solution, more precisely, all invertible $4\times 4$ matrices $R,Q$ that solve Yang--Baxter system related to quantised braided groups, quantum doubles and other systems are given.
Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum Yang-Baxter equation, or more properly the braid equation, is investigated in this setting. A solution of the braid equation in this…
The paper extends the notion of braided set and its close relative - the Yang-Baxter set - to the category of vector spaces and explore structure aspects of such a notion as morphisms and extensions. In this way we describe a family of…
We present a method to construct infinite families of entangling $2$-qudit gates, and amongst them entangling $2$-qudit gates which satisfy the Yang-Baxter equation. We show that, given $2$-qudit gates $c$ and $d$, if $c$ or $d$ is…
The representation theory of the Drinfeld doubles of dihedral groups is used to solve the Yang-Baxter equation. Use of the 2-dimensional representations recovers the six-vertex model solution. Solutions in arbitrary dimensions, which are…
We study noninvolutive set-theoretic solutions $(X,r)$ of the Yang-Baxter equations in terms of the properties of the canonically associated algebraic objects-the braided monoid $S(X,r)$, the quadratic Yang-Baxter algebra $A= A(\textbf{k},…
We use the method of the tensor product graph to construct rational (Yangian invariant) solutions of the Yang-Baxter equation in fundamental representations of $c_n$ and thence the full set of $c_n$-invariant factorized $S$-matrices. Brief…
In the first part we recall two famous sources of solutions to the Yang-Baxter equation -- R-matrices and Yetter-Drinfel$'$d (=YD) modules -- and an interpretation of the former as a particular case of the latter. We show that this result…
Let $V$ be a braided vector space, that is, a vector space together with a solution $\hat{R}\in {\text{End}}(V\otimes V)$ of the Yang--Baxter equation. Denote $T(V):=\bigoplus_k V^{\otimes k}$. We associate to $\hat{R}$ a solution…
The quantum Yang-Baxter equation is a braiding condition on vector spaces which is of high relevance in several fields of mathematics, such as knot theory and quantum group theory. Their combinatorial counterpart are set-theoretic solutions…
Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution consists of a set $X$ and a function r:X x X --> X x X which satisfies the braid relation. We…
We describe several methods of constructing R-matrices that are dependent upon many parameters, for example unitary R-matrices and R-matrices whose entries are functions. As an application, we construct examples of R-matrices with…
We consider the irreducible representations each of dimension 2 of the necklace braid group $\mathcal{NB}_n$ ($n=2,3,4$). We then consider the tensor product of the representations of $\mathcal{NB}_n$ ($n=2,3,4$) and determine necessary and…
We give a selfcontained introduction to the theory of quantum groups according to Drinfeld highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras,…
Using a theorem of Schechtman - Varchenko on integral expressions for solutions of Knizhnik - Zamolodchikov equations we prove that the solutions of the Yang - Baxter equation associated to complex simple Lie algebras belong to the class of…
These are lecture notes prepared for a minicourse given at the Cimpa Research School "Algebraic and geometric aspects of representation theory", held in Curitiba, Brazil in March 2013. The purpose of the course is to provide an introduction…