相关论文: On representations of braid groups determined by d…

The correspondence of the braid group on a handlebody of arbitrary genus to the algebra of Yang-Baxter and extended reflection equation operators is shown. Representations of the infinite dimensional extended reflection equation algebra in…

In this paper we indicate one method of construction of linear representations of groups and algebras with translation invariant (except, maybe , finite number) defining relationships. As an illustration of this method, we give one approach…

In this paper, we first demonstrate that a finite-dimensional $n$-Leibniz algebra naturally gives rise to an $n$-rack structure on the underlying vector space. Given any $n$-Leibniz algebra, we also construct two Yang-Baxter operators on…

Derived braids have been used to classify categorical structures based on the braid underlying a braided monoidal category V. With four-strand braids underlying the composition morphisms of tensor products of categories enriched over V,…

We study non-degenerate involutive set-theoretic solutions (X,r) of the Yang-Baxter equation, we call them simply solutions. We show that the structure group G(X,r) of a finite non-trivial solution (X,r) cannot be an Engel group. It is…

In braided tensor categories we show the Maschke's theorem and give the necessary and sufficient conditions for double cross biproducts and crossbiproducts and biproducts to be bialgebras. We obtain the factorization theorem for braided…

In this paper we discuss representations of the Birman-Wenzl-Murakami algebra as well as of its dilute extension containing several free parameters. These representations are based on superalgebras and their baxterizations permit us to…

We study indecomposable involutive set-theoretic solutions of the Yang-Baxter equation with cyclic permutation groups (cocyclic solutions). In particular, we show that there is no one-to-one correspondence between indecomposable cocyclic…

We compute the braided groups and braided matrices $B(R)$ for the solution $R$ of the Yang-Baxter equation associated to the quantum Heisenberg group. We also show that a particular extension of the quantum Heisenberg group is dual to the…

We introduce a novel algebraic structure called di-skew brace by which we show that generalized digroups systematically yield bijective, non-degenerate solutions to the set-theoretic Yang-Baxter equation. We study the structural properties…

We present particularly simple new solutions to the Yang--Baxter equation arising from two--dimensional cyclic representations of quantum $SU(2)$. They are readily interpreted as scattering matrices of relativistic objects, and the quantum…

Let k be a field. Let also (F, G) be a matched pair of groups. We give necessary and sufficient conditions on a pair (\sigma, \tau) of 2-cocycles in order that the crossed product algebra and the crossed coproduct coalgebra…

We present Yang-Baxter maps associated to elliptic curves. They are related to discrete versions of the Krichever-Novikov and the Landau-Lifshits equations. A lifting of scalar integrable quad-graph equations to two-field equations is also…

We show that the Yang-Baxter equation is equivalent to the associativity of the algebra generated by non-commuting link operators. Starting from these link operators we build out the (FFZ) algebras, the $s\ell_q (2)$ is derived by…

We introduce framed versions of the $L$-moves and prove a one move theorem for the extension of the Markov theorem for framed braids. We further introduce framed versions of the Hilden and Pure Hilden groups, we give presentations and we…

We give an algorithm to decide if a given braid is a product of two factors which are conjugates of given powers of standard generators of the braid group. The same problem is solved in a certain class of Garside groups including Artin-Tits…

We introduce non-degenerate solutions of the Yang-Baxter equation in the setting of symmetric monoidal categories. Our theory includes non-degenerate set-theoretical solutions as basic examples. However, infinite families of non-degenerate…

We study 2-reductive non-involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation. We give a combinatorial construction of any such solution of any (even infinite) size. We also prove that solutions associated to a skew…

Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. They are a class of algebras with triangular decomposition, arising from a deformation problem, the…

Yu. I. Merzljakov developed a method of splittable coordinates which helps to verify the linearity of some groups, he established some fundamental results using this method. In this paper we use the method of splittable coordinates and find…