Article
Mathematics, Applied
F. Cedo, E. Jespers, J. Okninski
Summary: In this paper, it is proven that every primitive permutation group is of prime order, and a specific construction method is provided for solutions of this type. This result is of great significance for the classification problem of all involutive non-degenerate set-theoretic solutions.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2022)
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, Applied
Marco Castelli
Summary: In this paper, we characterize finite simple involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation using left braces and provide some significant examples.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics
F. Cedo, J. Okninski
Summary: This study focuses on involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation on a finite set, with an emphasis on the case of indecomposable solutions. The research aims to determine how these solutions are built from imprimitivity blocks and characterize these blocks. Specifically, the study constructs several infinite families of simple solutions for the first time and completely characterizes a broad class of simple solutions of order p(2) for any prime p.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics, Applied
Francesco Catino, Ilaria Colazzo, Paola Stefanelli
Summary: This paper introduces a construction technique called strong semilattice of solutions for set-theoretic solutions of the Yang-Baxter equation, which allows one to obtain new solutions, in particular non-bijective solutions of finite order. It also explores a generalization of the algebraic structure of semi-braces based on this new construction technique of solutions, as braces, skew braces, and semi-braces are closely linked with solutions.
FORUM MATHEMATICUM
(2021)
Article
Mathematics, Applied
A. Ballester-Bolinches, R. Esteban-Romero, N. Fuster-Corral, H. Meng
Summary: The paper describes the relationship between the structure group, permutation group, and a set-theoretic solution of the quantum Yang-Baxter equation using the Cayley graph of the permutation group. By analyzing the left brace structure, new properties of the groups are obtained and known properties are recovered in a more transparent way.
MEDITERRANEAN JOURNAL OF MATHEMATICS
(2021)
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, Applied
Marco Trombetti
Summary: The aim of this paper is to prove that the structure skew brace associated with a finite non-degenerate solution of the Yang-Baxter equation is finitely presented.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics, Applied
Marco Castelli, Francesco Catino, Paola Stefanelli
Summary: The main aim of this paper is to provide sufficient conditions for left non-degenerate bijective set-theoretic solutions of the Yang-Baxter equation to be non-degenerate. Additionally, it extends previous results on involutive solutions and answers a question posed by Cedo et al. Furthermore, a theory of extensions is developed to construct new families of set-theoretic solutions.
JOURNAL OF ALGEBRA AND ITS APPLICATIONS
(2022)
Article
Mathematics
F. Cedo, J. Okninski
Summary: This article studies the indecomposable involutive non-degenerate set-theoretic solutions (X, r) of the Yang-Baxter equation with cardinality p1 & BULL; & BULL; & BULL; pn, where p1, . . . , pn are different prime numbers. It is proven that these solutions are multipermutation solutions of level < n. It also solves a problem stated in [11] and extends earlier results on indecomposability of solutions. The proofs are based on a detailed study of the brace structure on the permutation group c(X, r) associated with such a solution. Additionally, indecomposable solutions of cardinality p1 & BULL; & BULL; & BULL; pn that are multipermutation of level n are constructed for every nonnegative integer n.
ADVANCES IN MATHEMATICS
(2023)
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
Francesco Catino, Marzia Mazzotta, Paola Stefanelli
Summary: The paper aims to provide set-theoretical solutions of the Yang-Baxter equation, including new idempotent ones, by drawing on both classical theory of inverse semigroups and recently studied braces. It introduces a new structure, the inverse semi-brace, to offer new constructions allowing for obtaining new solutions of the Yang-Baxter equation.
JOURNAL OF ALGEBRA
(2021)
Article
Mathematics
F. Cedo, J. Okninski
Summary: This paper examines involutive non-degenerate set theoretic solutions of the Yang-Baxter equation, with a specific focus on finite solutions. The study identifies a rich class of indecomposable and irretractable solutions, as well as the necessary and sufficient conditions for these solutions to be simple. Additionally, the paper establishes a link between simple solutions and simple left braces, enabling the construction of more examples of simple solutions. Overall, the research addresses previous problems and presents new approaches.
JOURNAL OF ALGEBRA
(2022)
Article
Physics, Multidisciplinary
Anastasia Doikou
Summary: This study focuses on a type of solutions to the Yang-Baxter equation, which can be obtained using an algebraic structure called braces. The aim is to express these solutions in terms of admissible Drinfeld twists, extending recent findings. By identifying the generic form of the twists associated with set-theoretic solutions and proving their admissibility, this research also applies to Baxterized solutions of the YBE.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2021)
Article
Mathematics, Applied
F. Cedo, E. Jespers, L. Kubat, A. Van Antwerpen, C. Verwimp
Summary: This article investigates the structure of solutions for finite sets, characterizing the nilpotent property of the structure monoid under certain conditions. The results are applied to problems involving racks and multipermutations. Additionally, the article proves that the torsion part of a finite group is finite and related to the additive structure of the skew left brace.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2023)
Article
Mathematics, Applied
Ferran Cedo, Eric Jespers, Jan Okninski
Summary: This paper examines the structure K-algebra of a non-degenerate set-theoretic solution (X, r) and a field K, focusing on the dimension of A. The concept of derived solution is introduced to determine lower bounds and classify solutions based on these bounds, including the general case and the square-free case. Several problems posed by Gateva-Ivanova in 2018 are addressed in this study.
REVISTA MATEMATICA COMPLUTENSE
(2021)
Article
Mathematics, Applied
F. Cedo, E. Jespers, J. Okninski
Summary: Left braces, introduced by Rump, have been shown to be an important tool in studying set-theoretic solutions of the quantum Yang-Baxter equation, allowing for the construction of new families of solutions. The main result of the paper demonstrates that every finite abelian group is a subgroup of the additive group of a finite simple left brace with metabelian multiplicative group with abelian Sylow subgroups. This complements the authors' earlier unexpected results on a abundance of finite simple left braces.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2021)
Article
Mathematics
Ferran Cedo, Eric Jespers, Charlotte Verwimp
Summary: The paper discusses the structure and properties of set-theoretic solutions of the Yang-Baxter equation, focusing on 1-cocycles and their bijectivity. In the case of finite X, the relationship between the bijectivity of pi and pi' and the left and right non-degeneracy of (X, r) is established. Additionally, it is shown that non-degenerate irretractable solutions are necessarily bijective.
PUBLICACIONS MATEMATIQUES
(2021)
Article
Mathematics
F. Cedo, J. Okninski
Summary: This study focuses on involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation on a finite set, with an emphasis on the case of indecomposable solutions. The research aims to determine how these solutions are built from imprimitivity blocks and characterize these blocks. Specifically, the study constructs several infinite families of simple solutions for the first time and completely characterizes a broad class of simple solutions of order p(2) for any prime p.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics, Applied
F. Cedo, E. Jespers, J. Okninski
Summary: In this paper, it is proven that every primitive permutation group is of prime order, and a specific construction method is provided for solutions of this type. This result is of great significance for the classification problem of all involutive non-degenerate set-theoretic solutions.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2022)
Article
Mathematics
F. Cedo, J. Okninski
Summary: This paper examines involutive non-degenerate set theoretic solutions of the Yang-Baxter equation, with a specific focus on finite solutions. The study identifies a rich class of indecomposable and irretractable solutions, as well as the necessary and sufficient conditions for these solutions to be simple. Additionally, the paper establishes a link between simple solutions and simple left braces, enabling the construction of more examples of simple solutions. Overall, the research addresses previous problems and presents new approaches.
JOURNAL OF ALGEBRA
(2022)
Article
Mathematics
Andreas Baechle, Geoffrey Janssens, Eric Jespers, Ann Kiefer, Doryan Temmerman
Summary: The article investigates the properties of the unit group U(ZG) of the integral group ring ZG and proves that when G is a finite group satisfying certain conditions, U(ZG) either satisfies Kazhdan's property (T) or is a non-trivial amalgamated product. The proof involves the construction of amalgamated decompositions based on rational division algebra and arithmetic groups, and the application of these methods to higher dimensional modular and Bianchi groups.
JOURNAL OF ALGEBRA
(2022)
Article
Mathematics
Francesco Catino, Ferran Cedo, Paola Stefanelli
Summary: We introduce left and right series of left semi-braces and define left and right nilpotent left semi-braces. We study the structure of these semi-braces, generalize some results from skew left braces to left semi-braces, and analyze the cases where the set of additive idempotents is an ideal of the left semi-braces. Finally, we introduce the concept of nilpotent left semi-braces and prove that their multiplicative groups are nilpotent.
JOURNAL OF ALGEBRA
(2022)
Article
Mathematics, Applied
E. Jespers, A. Van Antwerpen, L. Vendramin
Summary: We study the relationships between different notions of nilpotency in skew braces and their applications to solving the Yang-Baxter equation. Specifically, we investigate annihilator nilpotent skew braces, an important class that can be thought of as analogous to nilpotent groups in the context of brace theory.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2023)
Article
Mathematics
I. Colazzo, E. Jespers, A. Van Antwerpen, C. Verwimp
Summary: The algebraic structure of YB-semitrusses is investigated, showing the connection between the right non-degeneracy and bijectivity of finite left non-degenerate set-theoretic solutions of the Yang-Baxter equation. It is also demonstrated that some finite left non-degenerate solutions can be reduced to non-degenerate solutions of smaller size.
JOURNAL OF ALGEBRA
(2022)
Review
Mathematics
Andreas Bachle, Geoffrey Janssens, Eric Jespers, Ann Kiefer, Doryan Temmerman
Summary: This paper proves a unit theorem characterizing the condition that the unit group U(ZG) of the integral group ring ZG satisfies Kazhdan's property (T), and shows that this property is equivalent to FAb and HFA properties. Moreover, it describes the simple epimorphic images of QG in groups with the cut property. The proof of the unit theorem relies on fixed point properties and the abelianization of elementary subgroups of SLn(D).
MATHEMATISCHE NACHRICHTEN
(2023)
Article
Mathematics, Applied
F. Cedo, E. Jespers, L. Kubat, A. Van Antwerpen, C. Verwimp
Summary: This article investigates the structure of solutions for finite sets, characterizing the nilpotent property of the structure monoid under certain conditions. The results are applied to problems involving racks and multipermutations. Additionally, the article proves that the torsion part of a finite group is finite and related to the additive structure of the skew left brace.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2023)
Article
Mathematics
E. Jespers
Summary: Significant progress has been made in constructing large torsion-free subgroups of the unit group of the integral group ring of a finite group, relying on explicit constructions of units and the description of Wedderburn components. The existence of reduced two degree representations plays a crucial role in these constructions, despite the unit group not being fully understood.
INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS
(2021)
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)
