Related papers: Braces and the Yang-Baxter equation
Braces were introduced by Rump to study involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation. A constructive method for producing all such finite solutions from a description of all finite left braces has been…
This paper aims to introduce a construction technique of set-theoretic solutions of the Yang-Baxter equation, called strong semilattice of solutions. This technique, inspired by the strong semilattice of semigroups, allows one to obtain new…
The homogeneous Yang-Baxter deformation is part of a larger web of integrable deformations and dualities that recently have been studied with motivations in integrable $\sigma$-models, solution-generating techniques in supergravity and…
Associative Yang-Baxter equation arises in different areas of algebra, e.g., when studying double quadratic Poisson brackets, non-abelian quadratic Poisson brackets, or associative algebras with cyclic 2-cocycle (anti-Frobenius algebras).…
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…
W. Rump showed that there exists a one-to-one correspondence between involutive right non-degenerate solutions of the Yang-Baxter equation and Rump right quasigroups. J. S. Carter, M. Elhamdadi, and M. Saito, meanwhile, introduced a…
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…
A connection between the Yang-Baxter relation for maps and the multi-dimensional consistency property of integrable equations on quad-graphs is investigated. The approach is based on the symmetry analysis of the corresponding equations. It…
Our primary focus is on the theory of skew braces, specifically exploring their connection with combinatorial solutions to the Yang-Baxter equation. Skew braces have recently emerged as intriguing algebraic structures, and their link to the…
The pioneering work of Rump, which proved Gateva-Ivanova's conjecture concerning the decomposability of square-free solutions to the Yang-Baxter equation, significantly motivated further research into the associated squaring map $T$. This…
This paper aims to deepen the theory of bijective non-degenerate set-theoretic solutions of the Yang-Baxter equation, not necessarily involutive, by means of q-cycle sets. We entirely focus on the finite indecomposable ones among which we…
Idempotent left nondegenerate solutions of the Yang-Baxter equation are in one-to-one correspondence with twisted Ward left quasigroups, which are left quasigroups satisfying the identity $(x*y)*(x*z)=(y*y)*(y*z)$. Using combinatorial…
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…
In this paper, we study the class of indecomposable involutive solutions of the Yang-Baxter equation of finite primitive level, recently introduced by Ced\'o and Okni\'nski in \cite{cedo2021constructing}. We give a group-theoretic…
Set-theoretic solutions to the Yang-Baxter equation have been studied extensively by means of related algebraic systems such as cycle sets and braces, dynamical versions of which have also been developed. No work focuses on set-theoretic…
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…
Given a right-non-degenerate set-theoretic solution $(X,r)$ to the Yang-Baxter equation, we construct a whole family of YBE solutions $r^{(k)}$ on $X$ indexed by its reflections $k$ (i.e., solutions to the reflection equation for $r$). This…
It is proven that finite idempotent left non-degenerate set-theoretic solutions $(X,r)$ of the Yang-Baxter equation on a set $X$ are determined by a left simple semigroup structure on $X$ (in particular, a finite union of isomorphic copies…
Recently V.Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested to consider ``set-theoretical'' solutions of the quantum Yang-Baxter equation, i.e. solutions given by a permutation $R$ of the set…
As generalizations of inverse semibraces introduced by Catino, Mazzotta and Stefanelli, Miccoli has introduced regular $\star$-semibraces under the name of involution semibraces and given a sufficient condition under which the associated…