Related papers: Qubit representations of the braid groups from gen…
In this note we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order)…
We investigate the braid group representations arising from categories of representations of twisted quantum doubles of finite groups. For these categories, we show that the resulting braid group representations always factor through finite…
By a generalized Yangian we mean a Yangian-like algebra of one of two classes. One of these classes consists of the so-called braided Yangians, introduced in our previous paper. The braided Yangians are in a sense similar to the reflection…
Generalization of the quantum Yang-Baxter equation solutions to an arbitrary grading is studied. The noncommutative differential calculi corresponding to such solutions is considered. The connection with the ordinary and supersymmetric…
We study combinatorial properties of virtual braid groups and we describe relations with finite type invariant theory for virtual knots and Yang-Baxter equations
We extend our recent work on set-theoretic solutions of the Yang-Baxter or braid relations with new results about their automorphism groups, strong twisted unions of solutions and multipermutation solutions. We introduce and study graphs of…
We describe the monodromy of dynamical Knizhnik-Zamolodchikov equations via Stokes phenomenon. It defines a family of braid groups representations by certain Stokes matrices. In particular, these Stokes matrices satisfy the Yang-Baxter…
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.
Yangian-like algebras, associated with current R-matrices, different from the Yang ones, are introduced. These algebras are of two types. The so-called braided Yangians are close to the Reflection Equation algebras, arising from involutive…
In this paper a class of new quantum groups is presented: deformed Yangians. They arise from rational solutions of the classical Yang-Baxter equation of the form $c_2 /u + const$ . The universal quantum $R$-matrix for a deformed Yangian is…
Modified braid equations satisfied by generalized ${\hat R}$ matrices (for a {\em given} set of group relations obeyed by the elements of ${\sf T}$ matrices ) are constructed for q-deformed quantum groups $GL_q (N), SO_q (N)$ and $Sp_q (N)$…
In this note, we adapt the procedure of the Long-Moody procedure to construct linear representations of welded braid groups. We exhibit the natural setting in this context and compute the first examples of representations we obtain thanks…
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…
We define and study extensions of Artin's representation and braid monodromy representation to the case of topological and algebraical generalisations of braid groups. In particular we provide faithful representations of braid groups of…
Quantum doubles of finite group algebras form a class of quasi-triangular Hopf algebras which algebraically solve the Yang--Baxter equation. Each representation of the quantum double then gives a matrix solution of the Yang--Baxter…
We establish a one-to-one correspondence between a class of Garside groups admitting a certain presentation and the structure groups of non-degenerate, involutive and braided set-theoretical solutions of the quantum Yang-Baxter equation. We…
The most common geometric interpretation of the Yang-Baxter equation is by braids, knots and relevant Reidemeister moves. So far, cubes were used for connections with the third Reidemeister move only. We will show that there are…
A unitary operator that satisfies the constant Yang-Baxter equation immediately yields a unitary representation of the braid group B n for every $n \ge 2$. If we view such an operator as a quantum-computational gate, then topological…
We study the Yang-Baxter equation for the $R$-matrices of the six-vertex model. We analyze the solutions and give new parametrizations of the Yang-Baxter equation. In particular, we find the maximal commutative families of parametrized…
We develop a general theory of `quantum' diffeomorphism groups based on the universal comeasuring quantum group $M(A)$ associated to an algebra $A$ and its various quotients. Explicit formulae are introduced for this construction, as well…