Related papers: Segre products and Segre morphisms in a class of Y…

We study the Yang-Baxter algebras $A(K,X,r)$ associated to finite set-theoretic solutions $(X,r)$ of the braid relations. We introduce an equivalent set of quadratic relations $\Re\subseteq G$, where $G$ is the reduced Gr\"obner basis of…

Given a finite non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation and a field $K$, the structure $K$-algebra of $(X,r)$ is $A=A(K,X,r)=K\langle X\mid xy=uv \mbox{ whenever }r(x,y)=(u,v)\rangle$. Note that…

We study $d$-Veronese subalgebras $A^{(d)}$ of Yang-Baxter algebras $A_X= A(K, X, r)$ related to finite nondegenerate involutive set-theoretic solutions $(X, r)$ of the Yang-Baxter equation, where $K$ is a field and $d\geq 2$ is an integer.…

We study the quadratic algebras $A(K,X,r)$ associated to a class of strictly braided but idempotent set-theoretic solutions $(X,r)$ of the Yang-Baxter or braid relations. In the invertible case, these algebras would be analogues of…

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…

A well-known noncommutative deformation $\mathcal A^N_{\mathbf{q}}$ of the polynomial algebra $\mathcal A^N$ can be obtained as a twist of $\mathcal A^N$ by a cocycle on the grading semigroup. Of particular interest to us is an…

In this paper we define and study convolution products for modules over certain families of VV algebras. We go on to study morphisms between these products which yield solutions to the Yang-Baxter equation so that in fact these morphisms…

We introduce the notion of the twisted Segre product $A\circ_\psi B$ of $\mathbb Z$-graded algebras $A$ and $B$ with respect to a twisting map $\psi$. It is proved that if $A$ and $B$ are noetherian Koszul Artin-Schelter regular algebras…

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},…

Inspired by the work [PA], we establish an explicit algebra isomorphism between the degenerate cyclotomic Yokonuma-Hecke algebra $Y_{r,n}^{d}(q)$ and a direct sum of matrix algebras over tensor products of degenerate cyclotomic Hecke…

A new method to construct involutive non-degenerate set-theoretic solutions $(X^n,r^{(n)})$ of the Yang-Baxter equation from an initial solution $(X,r)$ is given. Furthermore, the permutation group $\mathcal{G}(X^n,r^{(n)})$ associated to…

A bijective map $r: X^2 \longrightarrow X^2$, where $X = \{x_1, ..., x_n \}$ is a finite set, is called a \emph{set-theoretic solution of the Yang-Baxter equation} (YBE) if the braid relation $r_{12}r_{23}r_{12} = r_{23}r_{12}r_{23}$ holds…

We present resent results regarding invertible, non-degenerate solutions of the set-theoretic Yang-Baxter and reflection equations. We recall the notion of braces and we present and prove various fundamental properties required for the…

Yang-Baxter bialgebras, as previously introduced by the authors, are shown to arise from a double crossproduct construction applied to the bialgebra R T T = T T R, E T = T E R, \Delta(T) = T \hat{\otimes} T, \Delta(E) = E \hat{\otimes} T +…

Hyperplanes and hyperplane complements in the Segre product of partial linear spaces are investigated . The parallelism of such a complement is characterized in terms of the point-line incidence. Assumptions, under which the automorphisms…

Let $X \subset Y$ be closed (possibly singular) subschemes of a smooth projective toric variety $T$. We show how to compute the Segre class $s(X,Y)$ as a class in the Chow group of $T$. Building on this, we give effective methods to compute…

On a reduced analytic space $X$ we introduce the concept of a generalized cycle, which extends the notion of a formal sum of analytic subspaces to include also a form part. We then consider a suitable equivalence relation and corresponding…

We survey the matrix product solutions of the Yang-Baxter equation obtained recently from the tetrahedron equation. They form a family of quantum $R$ matrices of generalized quantum groups interpolating the symmetric tensor representations…

To every involutive non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation on a finite set $X$ there is a naturally associated finite solvable permutation group ${\mathcal G}(X,r)$ acting on $X$. We prove that every…

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…