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
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
Astronomy & Astrophysics
Ali Eghbali, Tayebe Parvizi, Adel Rezaei-Aghdam
Summary: This article generalizes the Yang-Baxter deformation of Wess-Zumino-Witten model to the Lie supergroups case. The classification of YB deformations of WZW models based on Lie supergroups is made possible by utilizing various solutions of the (modified) graded classical Yang-Baxter equation ((m)GCYBE). Inequivalent solutions of the (m)GCYBE for gl(1|1) and (C3 + A) Lie superalgebras are obtained using non-standard basis and automorphism transformations. The YB deformations of WZW models based on GL(1|1) and (C3 + A) Lie supergroups are specified by skew-supersymmetric classical r-matrices satisfying (m)GCYBE. Conformal invariance is confirmed for GL(1|1) and (C3 + A) WZW models within the classes of YB deformations preserving conformal invariance.
Article
Materials Science, Multidisciplinary
Ashfaque H. Bokhari, Tahir Hussain, Jamshed Khan, Uzma Nasib
Summary: This paper investigates homothetic vector fields of Bianchi type I spacetimes using Rif tree approach. By transforming the homothetic symmetry equations to the reduced involutive form, the problem is split into cases and solved with restrictions on metric functions, resulting in new physically realistic metrics.
RESULTS IN PHYSICS
(2021)
Article
Mathematics
Wolfgang Rump
Summary: Set-theoretic solutions to the Yang-Baxter equation of multipermutation level two are classified via transvection orbits which are abelian torsors. The relationship to square-free solutions and rack solutions is determined by using the square map and its relationship to non-degeneracy of the corresponding cycle set. The crucial role of left and right ideal powers of braces is discussed in connection with applications to higher multipermutation level.
ALGEBRAS AND REPRESENTATION THEORY
(2022)
Article
Mathematics, Applied
O. Akgun, M. Mereb, L. Vendramin
Summary: In this study, Constraint Satisfaction methods are employed to enumerate and construct set-theoretic solutions to the Yang-Baxter equation of small size. The results show the number of involutive and non-involutive solutions for different sizes, and the method is also utilized to enumerate non-involutive biquandles.
MATHEMATICS OF COMPUTATION
(2022)
Article
Mathematics
Wolfgang Rump
Summary: This article investigates the relationship between the prime factors dividing the size of a finite indecomposable set-theoretic solution and the order of the associated permutation group. It is proven that the prime sets are equal when the size is square-free. Additionally, the article characterizes equality and demonstrates that surjective morphisms can be factorized into a covering and a morphism given by a brace ideal. The existence of solutions with non-equality of the prime sets is also explored, and a minimal example is constructed.
BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY-SIMON STEVIN
(2023)
Article
Mathematics
Matteo Varbaro, Hongmiao Yu
Summary: In this paper, a liaison theory via quasi-Gorenstein varieties is developed, and it is applied to derive the connectedness property of general quasi-Gorenstein subspace arrangements and the classical topological Lefschetz duality.
JOURNAL OF ALGEBRA
(2024)
Article
Mathematics
Eric J. Hanson, Xinrui You
Summary: In this paper, we demonstrate the use of arcs in computing bases for the Hom-spaces and first extension spaces between bricks over preprojective algebras of type A. We also classify the weak exceptional sequences over these algebras using this description. Furthermore, we explain the connection between our results and a similar combinatorial model for exceptional sequences over hereditary algebras of type A.
JOURNAL OF ALGEBRA
(2024)
Article
Mathematics
Valery Lunts, Spela Spenko, Michel Van den Bergh
Summary: This article provides a brief review of the cohomological Hall algebra and K-theoretical Hall algebra associated with quivers. It shows a homomorphism between them in the case of symmetric quivers. Additionally, the equivalence of categories of graded modules is established.
JOURNAL OF ALGEBRA
(2024)
Article
Mathematics
Luc Guyot, Ihsen Yengui
Summary: In this article, it is discussed that for any integral domain R, if R is a Bezout domain of Krull dimension <= 1, then its localization ring R(X) is also a Bezout domain of Krull dimension <= 1. The generalization of this result is explored in different cases such as valuation domains and lexicographic monomial orders, and an example is given to show that this result does not hold in the irrational case.
JOURNAL OF ALGEBRA
(2024)
Article
Mathematics
Pedro L. del Angel, E. Javier Elizondo, Cristhian Garay, Felipe Zaldivar
Summary: In this paper, we study the Grassmannian space of 2-dimensional isotropic subspaces with a specific form and symmetry, and characterize its irreducible subvarieties using symplectic Coxeter matroids. We also provide a complete characterization of symplectic matroids of rank 2 that can be represented over C.
JOURNAL OF ALGEBRA
(2024)
Article
Mathematics
Ioannis Emmanouil, Ilias Kaperonis
Summary: In this paper, we study the role of K-absolutely pure complexes in the homotopy category and the pure derived category. We prove that K-abspure is the isomorphic closure and investigate the relationship between strongly fp-injective modules and K-absolutely pure complexes. Furthermore, we demonstrate that, under certain conditions, a K-absolutely pure complex of strongly fp-injective modules can be a K(PInj)-preenvelope containing an injective module complex.
JOURNAL OF ALGEBRA
(2024)
Article
Mathematics
Miroslav Ploscica, Friedrich Wehrung
Summary: This study investigates the lattice of principal ideals in Abelian L-groups and presents relevant results. These results have important applications in the representation of distributive lattices and homomorphisms, as well as in solving the MV spectrum problem.
JOURNAL OF ALGEBRA
(2024)
Article
Mathematics
Christian Garcia, Thaisa Tamusiunas
Summary: We present a Galois correspondence for K-beta-rings, where beta is an action of a finite groupoid on a unital ring R. We recover the correspondence given in [11] for finite groupoids acting on commutative rings.
JOURNAL OF ALGEBRA
(2024)
Article
Mathematics
Andrea Lucchini, Dhara Thakkar
Summary: This paper studies the problem of minimum generating set for finite groups. By testing whether subsets of the group can generate the group, the minimum generating set can be determined. It is proved that the number of these tests can be significantly reduced if the chief series of the group is known, and at most |G|13/5 subsets need to be tested. This implies that the minimum generating set problem for finite groups can be solved in polynomial time.
JOURNAL OF ALGEBRA
(2024)
Article
Mathematics
Ulrich Meierfrankenfeld, Chris Parker, Gernot Stroth
Summary: This paper investigates the local and global structural properties of finite groups. By studying certain properties of finite groups, we obtain important conclusions about subgroups and extend previous research.
JOURNAL OF ALGEBRA
(2024)
Article
Mathematics
Shezad Mohamed
Summary: We prove the existence of a version of Weil descent, or Weil restriction, in the category of D-algebras. This result is obtained under a mild assumption on the associated endomorphisms. As a consequence, we establish the existence of the Weil descent functor in the category of difference algebras.
JOURNAL OF ALGEBRA
(2024)
Article
Mathematics
Annalisa Conversano, Nicolas Monod
Summary: This study solves the problem of whether all Lie groups can be represented faithfully on a countable set by reducing it to the case of simple Lie groups. It provides a solution for all solvable Lie groups and Lie groups with a linear Levi component, proving that every amenable locally compact second countable group acts faithfully on a countable set.
JOURNAL OF ALGEBRA
(2024)
Article
Mathematics
Wesley Fussner, George Metcalfe
Summary: This paper investigates the transfer of algebraic properties between quasivarieties and their relatively finitely subdirectly irreducible members, and establishes equivalences for certain properties under certain conditions. Additionally, the paper studies special cases of quasivarieties and proves decidability for possessing these properties under certain conditions.
JOURNAL OF ALGEBRA
(2024)
Article
Mathematics
Hao Li, Antun Milas
Summary: We analyze the structure of Feigin-Stoyanovsky's principal subspaces of affine Lie algebra and provide novel fermionic character formulas. We show that level one principal subspaces of type A are classically free as vertex algebras.
JOURNAL OF ALGEBRA
(2024)
Article
Mathematics
Raphael Ruimy
Summary: This article investigates the effect of the perverse t-structure in different dimensions and provides concrete examples. In the case of dimensions less than 2, the core of the t-structure is described. For schemes of finite type over a finite field, a best approximation of the perverse t-structure is constructed.
JOURNAL OF ALGEBRA
(2024)
