Article
Mathematics, Applied
H. Meng, A. Ballester-Bolinches, R. Esteban-Romero, N. Fuster-Corral
Summary: The text introduces new sufficient conditions for a group that can be factorised as a product of two IYB-groups to be an IYB-group, and shows that some earlier results are direct consequences of their main theorem.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
Wolfgang Rump
Summary: This study explores degenerate solutions to the Yang-Baxter equation using associated semibraces and groups. It separates a non-degenerate part from a purely degenerate one based on a characterization in terms of cycle sets, with the conclusion that every nontrivial Garside group leads to a degenerate cycle set. By employing a graded algebra related to the first Weyl algebra, a negative answer to a recent problem posed by Bonatto et al. (2021) is obtained.
JOURNAL OF ALGEBRA
(2022)
Article
Mathematics
Marco Bonatto, Michael Kinyon, David Stanovsky, Petr Vojtechovsky
Summary: Wolfgang Rump demonstrated the relationship between nondegenerate involutive set-theoretic solutions of the Yang-Baxter equation and binary algebras. Latin rumples are a focus, with specific conditions for the existence of affine solutions. A large class of affine solutions can be obtained from nonsingular near-circulant matrices.
JOURNAL OF ALGEBRA
(2021)
Article
Mathematics
Valeriy G. Bardakov, Vsevolod Gubarev
Summary: This article introduces the concepts of braces, skew left braces, and Rota-Baxter operators on groups, and establishes their connections.
JOURNAL OF ALGEBRA
(2022)
Article
Mathematics
A. Ballester-Bolinches, R. Esteban-Romero, P. Jimenez-Seral, V. Perez-Calabuig
Summary: This paper introduces and studies Yang-Baxter groups associated with not necessarily involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation. It provides sufficient conditions for a group that can be factorized as a product of two YB-groups to be a YB-group. Additionally, earlier results for finite IYB-groups are generalized for arbitrary (non-necessarily finite) YB-groups as a consequence of the main theorem presented.
QUAESTIONES MATHEMATICAE
(2023)
Article
Mathematics
S. Ramirez
Summary: This paper investigates the classification of indecomposable solutions of the Yang-Baxter equation. By using a proposed scheme by Bachiller, Cedo, and Jespers and recent advancements in the classification of braces, the authors classify all indecomposable solutions with certain permutation groups, including all groups of size pq, all abelian groups of size p(2)q, and all dihedral groups of size p(2)q.
COMMUNICATIONS IN ALGEBRA
(2023)
Article
Mathematics, Applied
Marco Castelli, Francesco Catino, Paola Stefanelli
Summary: This study examines a class of indecomposable involutive set-theoretic solutions of the Yang-Baxter equation with specific imprimitivity blocks, using the algebraic structure of left braces and the dynamical extensions of cycle sets. It also investigates one-generator left braces of multipermutation level 2.
MEDITERRANEAN JOURNAL OF MATHEMATICS
(2021)
Article
Physics, Multidisciplinary
Jon Links
Summary: The paper discusses the statement that every Yang-Baxter integrable system is exactly-solvable, introducing definitions and axioms to formalize it. A paradox is shown to arise in a specific Yang-Baxter integrable bosonic system, leading to the development of a generalization for completely integrable bosonic systems.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2021)
Article
Mathematics
E. Jespers, L. Kubat, A. Van Antwerpen, L. Vendramin
Summary: In this study, the radical and weight of a skew left brace are defined, along with some basic properties. A Wedderburn type decomposition for Artinian skew left braces is obtained, and analogues of a theorem of Wiegold, a theorem of Schur, and its converse in the context of skew left braces are proven. Finally, these results are applied to detect torsion in the structure group of a finite bijective non-degenerate set-theoretic solution of the Yang-Baxter equation.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics, Applied
Premysl Jedlicka, Agata Pilitowska, Anna Zamojska-Dzienio
Summary: The paper presents a construction of all finite indecomposable involutive solutions of the Yang-Baxter equation of multipermutational level at most 2 with an abelian permutation group. It derives a formula for the number of such solutions with a fixed number of elements and describes some properties of the automorphism groups, showing they are regular abelian groups in this case.
FORUM MATHEMATICUM
(2021)
Article
Mathematics
Premysl Jedlicka, Agata Pilitowska
Summary: We provide a complete characterization of all indecomposable involutive solutions to the Yang-Baxter equation at multipermutation level 2. Firstly, we construct a family of such solutions. Then, we prove that every indecomposable involutive solution to the Yang-Baxter equation with multipermutation level 2 is a homomorphic image of a previously constructed solution. By analyzing this epimorphism, we are able to obtain all such solutions up to isomorphism and enumerate those of small sizes.
JOURNAL OF COMBINATORIAL THEORY SERIES A
(2023)
Article
Physics, Multidisciplinary
S. Igonin, V Kolesov, S. Konstantinou-Rizos, M. M. Preobrazhenskaia
Summary: The study focuses on tetrahedron maps and Yang-Baxter maps, clarifying the structure of the nonlinear algebraic relations that define linear tetrahedron maps, presenting transformations to obtain new maps, proving that the differential of a tetrahedron map on a manifold is also a tetrahedron map, and providing new examples of parametric Yang-Baxter and tetrahedron maps. The research also includes invariants for a nonlinear tetrahedron map and constructions of linear tetrahedron maps that serve as approximations for a nonlinear tetrahedron map in the study of soliton solutions.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2021)
Article
Mathematics, Applied
Andrew P. Kels
Summary: This article demonstrates how Yang-Baxter maps can be directly obtained from classical counterparts of the star-triangle relations and quantum Yang-Baxter equations. It is based on reinterpreting the latter equation and its solutions, given in terms of special functions, as a set-theoretical form of the Yang-Baxter equation, yielding quadrirational Yang-Baxter maps. The obtained Yang-Baxter maps satisfy two different types of Yang-Baxter equations, one involving a single map and the other involving a pair of maps, known as an entwining Yang-Baxter equation. Only the maps solving the former type of equation are reversible. Sixteen different Yang-Baxter maps are derived from known solutions of the classical star-triangle relations.
PHYSICA D-NONLINEAR PHENOMENA
(2023)
Article
Mathematics, Applied
Wolfgang Rump
Summary: The paper systematically studies involutive non-degenerate set-theoretic solutions to the Yang-Baxter equation, focusing on solutions with cyclic permutation groups.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
Shuangjian Guo, Shengxiang Wang, Xiaohui Zhang
Summary: In this paper, the concept of Hom-Leibniz bialgebras is introduced, which is equivalent to matched pairs of Hom-Leibniz algebras and Manin triples of Hom-Leibniz algebras. Additionally, the notion of relative Rota-Baxter operators is extended to Hom-Leibniz algebras, and it is proven that there is a Hom-pre-Leibniz algebra structure on Hom-Leibniz algebras with a relative Rota-Baxter operator. Finally, the classical Hom-Leibniz Yang-Baxter equation on Hom-Leibniz algebras is studied and its connection with the relative Rota-Baxter operator is presented.
