Related papers: The algebraic structure of left semi-trusses
The problem of constructing all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation recently has been reduced to the problem of describing all the left braces. In particular, the classification of all finite…
Braces are generalizations of radical rings, introduced by Rump to study involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation (YBE). Skew braces were also recently introduced as a tool to study not necessarily…
The study of non-degenerate set-theoretic solutions of the Yang-Baxter equation calls for a deep understanding of the algebraic structure of a skew left brace. In this paper, the skew brace theoretical property of solubility is introduced…
Semi-entwining structures are proposed as concepts simpler than entwining structures, yet they are shown to have interesting applications in constructing intertwining operators and braided algebras, lifting functors, finding solutions for…
We study classical twists of Lie bialgebra structures on the polynomial current algebra $\mathfrak{g}[u]$, where $\mathfrak{g}$ is a simple complex finite-dimensional Lie algebra. We focus on the structures induced by the so-called…
Pre-Novikov algebras and quasi-Frobenius Novikov algebras naturally appear in the theory of Novikov bialgebras. In this paper, we show that there is a natural pre-Novikov algebra structure associated to a quasi-Frobenius Novikov algebra.…
This paper introduces the tensor representation of a network, here tensors are the primitive structures of the network. In view of tensor chains, two binary operations on tensor sets are defined: chain addition and reducing. Based on the…
In this paper, the different operator forms of classical Yang-Baxter equation are given in the tensor expression through a unified algebraic method. It is closely related to left-symmetric algebras which play an important role in many…
This paper introduces Hopf braces, a new algebraic structure related to the Yang-Baxter equation which include Rump's braces and their non-commutative generalizations as particular cases. Several results of classical braces are still valid…
In this paper, we introduce the theory of Rota-Baxter operators on Clifford semigroups, useful tools for obtaining dual weak braces, i.e., triples $\left(S,+,\circ\right)$ where $\left(S,+\right)$ and $\left(S,\circ\right)$ are Clifford…
A skew brace, as introduced by L. Guarnieri and L. Vendramin, is a set with two group structures interacting in a particular way. When one of the group structures is abelian, one gets back the notion of brace as introduced by W. Rump. Skew…
Non-associtive algebras is a research direction gaining much attention these days. New developments show that associative algebras and some not-associative structures can be unified at the level of Yang-Baxter structures. In this paper, we…
We study solutions of the parametric set-theoretic reflection equation from an algebraic perspective by employing recently derived generalizations of the familiar shelves and racks, called parametric (p)-shelves and racks. Generic…
Cycle sets are algebraic structures introduced by Rump to study set theoretic solutions to the Yang-Baxter equation. While studying cycle sets Rump also introduced braces, which have since overtaken cycle sets as a tool for studying…
We consider involutive, non-degenerate, finite set theoretic solutions of the Yang-Baxter equation. Such solutions can be always obtained using certain algebraic structures that generalize nil potent rings called braces. Our main aim here…
As generalizations of Rota--Baxter groups, Rota--Baxter Clifford semigroups have been introduced by Catino, Mazzotta and Stefanelli in 2023. Based on their pioneering results, in this paper we first continue to study Rota--Baxter Clifford…
Given an associative multiplication in matrix algebra compatible with the usual one or, in other words, linear deformation of matrix algebra, we construct a solution to the classical Yang-Baxter equation. We also develop a theory of such…
In this paper, we propose a generalization for the class of laura algebras, which we call almost laura. We show that this new class of algebras retains most of the essential features of laura algebras, especially concerning the important…
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…
The Temperley-Lieb and Brauer algebras and their cyclotomic analogues, as well as the partition algebra, are all examples of twisted semigroup algebras. We prove a general theorem about the cellularity of twisted semigroup algebras of…