Related papers: Rota-Baxter groups, skew left braces, and the Yang…
For a skew left brace $(G, \cdot, \circ)$, the map $\lambda : (G, \circ) \to \Aut \,(G, \cdot),~~a \mapsto \lambda_a,$ where $\lambda_a(b) = a^{-1} \cdot (a \circ b)$ for all $a, b \in G$, is a group homomorphism. Then $\lambda$ can also be…
We study the class of one-generator solutions to the Yang-Baxter equation, extending some recent results concerning the classes of involutive and multipermutation solutions. Moreover we show the precise relationship between indecomposable…
An operator deformed quantum algebra is discovered exploiting the quantum Yang-Baxter equation with trigonometric R-matrix. This novel Hopf algebra along with its $q \to 1$ limit appear to be the most general Yang-Baxter algebra underlying…
Building on a result by W. Rump, we show how to exploit the right-cyclic law (x.y).(x.z) = (y.x).(y.z) in order to investigate the structure groups and monoids attached with (involutive nondegenerate) set-theoretic solutions of the…
The theory of the set-theoretic Yang-Baxter equation is reviewed from a purely algebraic point of view. We recall certain algebraic structures called shelves, racks and quandles. These objects satisfy a self-distributivity condition and…
Let $r:X^{2}\rightarrow X^{2}$ be a set-theoretic solution of the Yang-Baxter equation on a finite set $X$. It was proven by Gateva-Ivanova and Van den Bergh that if $r$ is non-degenerate and involutive then the algebra $K\langle x \in X…
At the previous congress (CRM 6), we reviewed the construction of Yang-Baxter operators from associative algebras, and presented some (colored) bialgebras and Yang-Baxter systems related to them. The current talk deals with Yang-Baxter…
Given a left brace $G$, a method to construct all the involutive, non-degenerate set-theoretic solutions $(Y,s)$ of the YBE, such that $\mathcal{G}(Y,s)\cong G$ is given. This method depends entirely on the brace structure of $G$.
The Rota-Baxter operator and the modified Rota-Baxter operator on various algebras are both important in mathematics and mathematical physics. The former is originated from the integration-by-parts formula and probability with applications…
Rota-Baxter systems of T. Brzezi\'{n}ski are a generalization of Rota-Baxter operators that are related to dendriform structures, associative Yang-Baxter pairs and covariant bialgebras. In this paper, we consider Rota-Baxter systems in the…
In this paper we define a class of braces, that we call module braces or $R$-braces, which are braces for which the additive group has also a module structure over a ring $R$, and for which the values of the gamma functions are…
This is a self-contained review on the theory of quantum group and its applications to modern physics. A brief introduction is given to the Yang-Baxter equation in integrable quantum field theory and lattice statistical physics. The quantum…
A modified Rota-Baxter algebra is an algebra equipped with an operator that satisfies the modified Yang-Baxter equation. In this paper, we define the cohomology of a modified Rota-Baxter algebra with coefficients in a suitable bimodule. We…
In this paper, using extension theory and cohomological approach we introduce the notion of the obstruction class for an inner post-Lie algebra being induced by a Rota-Baxter operator, and show that an inner post-Lie algebra is induced by a…
The homogeneous Rota-Baxter operators on the Witt and Virasoro algebras are classified. As applications, the induced solutions of the classical Yang-Baxter equation and the induced pre-Lie and PostLie algebra structures are obtained.
It is shown that over an arbitrary field there exists a nil algebra $R$ whose adjoint group $R^{o}$ is not an Engel group. This answers a question by Amberg and Sysak from 1997 [5] and answers related questions from [3, 44]. The case of an…
We define a concept of Hecke algebra for structure groups of set-theoretical solutions to the Yang--Baxter equation. As a comparison to Artin--Tits groups of spherical type, we study some properties of this construction, while also…
The aim of this article is to advance the knowledge on the theory of skew left braces. We introduce a subclass of skew left braces, which we denote by $\mathcal{I}_n$, $n \ge 1$, such that elements of the annihilator and lower central…
A new type of algebras that represent a generalization of both quantum groups and braided groups is defined. These algebras are given by a pair of solutions of the Yang--Baxter equation that satisfy some additional conditions. Several…
A dual weak brace is an algebraic structure $\left(S,\,+,\,\circ\right)$ including skew braces and giving rise to a set-theoretic solution of the Yang-Baxter equation. We show that such a map belongs to a family of set-theoretic solutions,…