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
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
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, 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, Applied
Marco Castelli, Marzia Mazzotta, Paola Stefanelli
Summary: This paper aims to deepen the theory of bijective non-degenerate set-theoretic solutions of the Yang-Baxter equation, not necessarily involutive, using q-cycle sets. We primarily focus on the class of finite indecomposable solutions, particularly studying simple solutions. We provide a group-theoretic characterization of these solutions, including their permutation groups, and discuss some unresolved questions.
FORUM MATHEMATICUM
(2022)
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
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)
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
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
Physics, Mathematical
Anastasia Doikou, Alexandros Ghionis, Bart Vlaar
Summary: This paper examines classes of quantum algebras and their q-analogues produced from involutive and non-degenerate set-theoretic solutions of the Yang-Baxter equation. The authors provide universal results on quasi-bialgebras and admissible Drinfeld twists, and show that the quantum algebras produced from set-theoretic solutions and their q-analogues are quasi-triangular quasi-bialgebras. They also construct admissible Drinfeld twists in the q-deformed case, subject to certain extra constraints dictated by the q-deformation. These findings greatly generalize recent relevant results and provide specific illustrative examples.
LETTERS IN MATHEMATICAL PHYSICS
(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
Wolfgang Rump
Summary: This study reveals the equivalence between non-degenerate cycle sets and non-degenerate set-theoretic solutions to the Yang Baxter equation. It shows that retractable primitive cycle sets belong to a small list found previously, while irretractable primitive torsion cycle sets generate a canonical brace with certain properties. The brace has a unique minimal non-zero ideal and a cyclic quotient brace, and the adjoint group of a specific ideal has a trivial center.
JOURNAL OF ALGEBRA
(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, 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
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
I Heckenberger, L. Vendramin
ALGEBRAS AND REPRESENTATION THEORY
(2019)
Article
Mathematics
Alexander Konovalov, Agata Smoktunowicz, Leandro Vendramin
Summary: This article introduces combinatorial representations of finite skew braces and discusses different concepts in the theory of skew braces using a database of small skew braces.
EXPERIMENTAL MATHEMATICS
(2021)
Article
Mathematics
Victoria Lebed, Leandro Vendramin
PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY
(2019)
Article
Mathematics
E. Acri, R. Lutowski, L. Vendramin
INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION
(2020)
Article
Mathematics
E. Jespers, L. Kubat, A. Van Antwerpen, L. Vendramin
MATHEMATISCHE ANNALEN
(2019)
Article
Mathematics
Agata Smoktunowicz, Leandro Vendramin, Robert Weston
JOURNAL OF ALGEBRA
(2020)
Article
Mathematics
Victoria Lebed, Leandro Vendramin
Summary: Given a right-non-degenerate set-theoretic solution (X, r) to the Yang-Baxter equation, we construct a family of YBE solutions r((k)) on X indexed by its reflections k. These solutions induce isomorphic actions of the braid group/monoid on X-n. The structure monoids of r and r((k)) are related by a bijective 1-cocycle-like map. We study the reflection equation for non-degenerate involutive YBE solutions and show its equivalence to simpler relations, providing systematic ways of constructing new reflections.
JOURNAL OF ALGEBRA
(2022)
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
S. Ramirez, L. Vendramin
Summary: Based on the cycle structure of a certain permutation associated with the solution, several other decomposability theorems are presented.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(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, 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, Applied
C. Dietzel, P. Menchon, L. Vendramin
Summary: We use Constraint Satisfaction Methods to construct and enumerate finite L-algebras up to isomorphism, which have recently been introduced by Rump and have applications in Garside theory, algebraic logic, and the study of the combinatorial Yang-Baxter equation. The database suggests the existence of bijections between certain classes of L-algebras and well-known combinatorial objects. We prove that Bell numbers enumerate isomorphism classes of finite linear L-algebras, and that finite regular L-algebras are in bijective correspondence with infinite-dimensional Young diagrams.
MATHEMATICS OF COMPUTATION
(2023)
Article
Mathematics
T. Letourmy, L. Vendramin
Summary: In this paper, we define isoclinism of skew braces and explore its various applications. We examine some properties of skew braces that remain invariant under isoclinism, such as right nilpotency. This result has implications in the theory of set-theoretic solutions to the Yang-Baxter equation. Additionally, we introduce isoclinic solutions and study multipermutation solutions under isoclinism.
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY
(2023)
Article
Mathematics
Leandro Vendramin
ADVANCES IN GROUP THEORY AND APPLICATIONS
(2019)
Article
Mathematics
Victoria Lebed, Leandro Vendramin
Summary: In this article, solutions to the Yang-Baxter equation and the reflection equation were studied. By constructing a series of YBE solutions, it was found that reflections can be used as a tool for studying YBE solutions.
JOURNAL OF ALGEBRA
(2022)
Article
Mathematics, Applied
Manoj K. Keshari, Sampat Sharma
Summary: Assuming R is an affine algebra of dimension d > 4 over a perfect field k of char = 2 and I is an ideal of R. (1) M Sd+1(R) is uniquely divisible prime to char k if R is reduced and k is infinite with c.d.(k) < 1. (2) Umd+1(R, I)/Ed+1(R, I) has a nice group structure if c.d.2(k) < 2. (3) Umd(R, I)/Ed(R, I) has a nice group structure if k is algebraically closed of char k = 2, 3 and either (i) k = Fp or (ii) R is normal.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Mathieu Anel, Georg Biedermann, Eric Finster, Andre Joyal
Summary: In this article, the work of Toen-Vezzosi and Lurie on Grothendieck topologies is revisited using the new tools of acyclic classes and congruences. An extended Grothendieck topology on any 8-topos is introduced and it is proven that the poset of extended Grothendieck topologies is isomorphic to that of topological localizations, hypercomplete localizations, Lawvere-Tierney topologies, and covering topologies. The notions of cotopological morphism, hypercompletion, hyperdescent, hypercoverings, hypersheaves, and forcing are also discussed.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Daniel Dugger, Christy Hazel, Clover May
Summary: This article provides a complete description of the derived category of perfect complexes of modules over the constant Mackey ring Z/$ for the cyclic group C2. While it is simple for $ odd, it relies on a new splitting theorem when $ = 2. The splitting theorem also allows for computing the associated Picard group and Balmer spectrum for compact objects in the derived category. Additionally, it gives a complete classification of finite modules over the C2-equivariant Eilenberg-MacLane spectrum HZ/2 and provides new proofs for some facts about RO(C2)-graded Bredon cohomology.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Mikhailo Dokuchaev, Itailma Rocha
Summary: In this study, we construct an abelian group C(Theta/R) formed by the isomorphism classes of partial generalized crossed products related to a unital partial representation Theta of a group G into the Picard semigroup PicS(R) of a non-necessarily commutative unital ring R. We identify an appropriate second partial cohomology group of G with a naturally defined subgroup C0(Theta/R) of C(Theta/R). Using these results, we generalize the works by Kanzaki and Miyashita by giving an analogue of the Chase-Harrison-Rosenberg exact sequence associated with an extension of rings and a unital partial representation of an arbitrary group into the monoid of R-subbimodules.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Markus Thuresson
Summary: Hereditary algebras are quasi-hereditary and exhibit certain regularity properties with respect to adapted partial orders. This article investigates the Ext-algebra of standard modules over path algebras of linear quivers and provides necessary and sufficient conditions for regular exact Borel subalgebras. The findings have implications for the understanding of linear quivers with arbitrary orientations.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Cordian Riener, Robin Schabert
Summary: This article focuses on the geometry of a class of hyperbolic polynomial families determined by linear conditions on the coefficients. These polynomials have all their roots on the real line. The set of hyperbolic polynomials is stratified according to the multiplicities of the real zeros, and this stratification also applies to the hyperbolic slices. The study shows that the local extreme points of hyperbolic slices correspond to hyperbolic polynomials with at most k distinct roots, and that the convex hull of such a family is generally a polyhedron. The article also explores the implications of these results for symmetric real varieties and symmetric semi-algebraic sets, particularly in terms of sparse representations and sampling.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Edward L. Green, Sibylle Schroll
Summary: This paper studies the ideal C in the path algebra KQ, proving that KQ/C is always finite dimensional with finite global dimension, and it is Morita equivalent to an incidence algebra.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Alexei Entin, Noam Pirani
Summary: This paper proves the existence of a Galois extension with ramification only at infinity for symmetric and alternating groups over finite fields of odd characteristic.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Yves Baudelaire Fomatati
Summary: This paper improves the algorithm for matrix factorization of polynomials, obtaining better results by refining the construction of one of the main ingredients of the algorithm.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Ippei Nagamachi, Teppei Takamatsu
Summary: In this paper, we study the invariants and related phenomena of regular varieties and rings over imperfect fields. We give a criterion for geometric normality of such rings, study the Picard schemes of curves, and define new invariants relating to δ-invariants, genus changes, conductors, and Jacobian numbers. As an application, we refine Tate's genus change theorem and show that the Jacobian number of a curve is 2p/(p - 1) times the genus change.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Zongzhu Lin, Li Qiao
Summary: This article studies the Rota-Baxter algebra structure on the field A = k((t)), with P being the projection map. The representation theory and regular-singular decompositions of finite dimensional A-vector spaces are examined. The main result shows that the category of finite dimensional representations is semisimple, consisting of three isomorphism classes of one-dimensional irreducible representations. Additionally, the article uses the result to compute the generalized class number. (c) 2023 Elsevier B.V. All rights reserved.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Stephen Lack, Giacomo Tendas
Summary: In this paper, we characterize accessible V-categories with limits of a specified class by introducing the notion of companion C for a class of weights & psi;. We then characterize these categories as accessibly embedded and C-virtually reflective in a presheaf V-category, as well as the V-categories of C-models of sketches. Our theorem extends to the case of any weakly sound class & psi; and provides a new perspective on weakly locally presentable categories.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Pradeep K. Rai
Summary: In 1956, Green provided a bound on the order of the Schur multiplier of p-groups. This bound, which depends on the order of the group, is the best possible. Over time, the bound has been improved by incorporating additional factors such as the minimal number of generators and the order of the derived subgroup. We further enhance these bounds by considering the group's nilpotency class, with special emphasis on the cases of class 2 and maximal class.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Benjamin Dequene
Summary: Gentle algebras are a class of finite-dimensional algebras introduced by I. Assem and A. Skowronski in the 1980s. Modules over such algebras can be described using string and band combinatorics in the associated gentle quiver, as studied by M.C.R. Butler and C.M. Ringel. Nilpotent endomorphisms of quiver representations induce linear transformations over vector spaces at each vertex. Among all nilpotent endomorphisms, a well-defined Jordan form exists for these representations. This paper focuses on subcategories generated by the indecomposable representations of a gentle quiver, including a fixed vertex in their support, and characterizes the vertices such that the objects of this subcategory are determined up to isomorphism by their generic Jordan form.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Mark Lawson, Aidan Sims, Alina Vdovina
Summary: We construct a family of groups that are higher dimensional generalizations of the Thompson groups using suitable higher rank graphs. Inspired by the K-theory of C*-algebras, we introduce group invariants and demonstrate that many of our groups are non-isomorphic to the Brin-Thompson groups nV, where n ≥ 2.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
