37 pages, 20 tables. We split a previous work arXiv:1912.11889 improving results and adding a treatment for the even case. v2: v2: arXiv identifier of our preprint on skew braces of size $p^2q$ of abelian type now included. v3: 43 pages, 20 tables, final version with significant improvements, to appear in Journal of Algebra and Its Applications
发表信息：
Journal of Algebra and Its Applications (2022), 21(03):2250062
In this paper we enumerate the skew braces of non-abelian type of size $p^2q$ for $p,q$ primes with $q>2$ by the classification of regular subgroups of the holomorph of the non-abelian groups of the same order. Since Crespo dealt with the case $q=2$, this paper completes the enumeration of skew braces of size $p^2q$ started in a previous work by the authors. In some cases, we provide also a structural description of the skew braces. As an application, we prove a conjecture posed by V. Bardakov, M. Neshchadim and M. Yadav.
L. N. Childs defined a bi-skew brace to be a skew brace such that if we swap the role of the two operations, then we find again a skew brace. In this paper, we give a systematic analysis of bi-skew braces. We study nilpotency and solubility, and connections between bi-skew braces and set-theoretic solutions of the Yang--Baxter equation. Further, we deal with Byott's conjecture in the case of bi-skew braces, and we use bi-skew braces as a tool to solve a classification problem proposed by L. Vendramin. In the final part, we investigate brace blocks, defined by A. Koch to be families of group op…
We introduce strong left ideals of skew braces and prove that they produce non-trivial decomposition of set-theoretic solutions of the Yang-Baxter equation. We study factorization of skew left braces through strong left ideals and we prove analogs of Itô's theorem in the context of skew left braces. As a corollary, we obtain applications to the retractability problem of involutive non-degenerate solutions of the Yang-Baxter equation. Finally, we classify skew braces that contain no non-trivial proper ideals.
We study series of left ideals of skew left braces that are analogs of upper central series of groups. These concepts allow us to define left and right nilpotent skew left braces. Several results related to these concepts are proved and applications to infinite left braces are given. Indecomposable solutions of the Yang-Baxter equation are explored using the structure of skew left braces.
Braces were introduced by Rump as a generalization of Jacobson radical rings. It turns out that braces allow us to use ring-theoretic and group-theoretic methods for studying involutive solutions to the Yang-Baxter equation. If braces are replaced by skew braces, then one can use similar methods for studying not necessarily involutive solutions. Here we collect problems related to (skew) braces and set-theoretic solutions to the Yang-Baxter equation.
We introduce the notion of Drinfeld twists for both set-theoretical YBE solutions and skew braces. We give examples of such twists and show that all twists between skew braces come from families of isomorphisms between their additive groups. We then describe the relation between these definitions and co-twists on FRT-type Hopf algebras in the category $\mathrm{SupLat}$, and prove that any co-twist on a co-quasitriangular Hopf algebra in $\mathrm{SupLat}$ induces a Drinfeld twist on its remnant skew brace. We go on to classify co-twists on bicrossproduct Hopf algebras coming from groups with un…
We construct all skew braces of size $pq$ (where $p>q$ are primes) by using Byott's classification of Hopf--Galois extensions of the same degree. For $p\not\equiv 1 \pmod{q}$ there exists only one skew brace which is the trivial one. When $p\equiv 1 \pmod{q}$, we have $2q+2$ skew braces, two of which are of cyclic type (so, contained in Rump's classification) and $2q$ of non-abelian type.
We define combinatorial representations of finite skew braces and use this idea to produce a database of skew braces of small size. This database is then used to explore different concepts of the theory of skew braces such as ideals, series of ideals, prime and semiprime ideals, Baer and Wedderburn radicals and solvability. The paper contains several questions.
The second cohomology group of a left skew brace with coefficients in a trivial left brace with non-trivial actions is defined, its connection with extensions of a left skew brace by a trivial braces is established and a Wells' like exact sequence relating the second cohomology group with inducible automorphisms of an extension of left skew braces is constructed.
In this paper we enumerate the skew braces of size $p^2q$ for $p,q$ odd primes by the classification of regular subgroups of the holomorph of the groups of size $p^2q$. In particular, we provide explicit formulas for the skew braces of abelian type.
We study relations between the additive and the multiplicative groups of a two-sided skew brace. In particular, we prove that if the additive group of a two-sided skew brace is finite solvable (respectively, finitely generated nilpotent, finitely generated residually nilpotent, finitely generated residually finite), then the multiplicative group of this skew brace is solvable (respectively, solvable, residually solvable, residually finite). Also, we prove that if the multiplicative group of a two-sided skew brace is nilpotent of nilpotency class $k$, then the additive group of this skew brace …
L. Childs has defined a skew brace $(G, \cdot, \circ)$ to be a bi-skew brace if $(G, \circ, \cdot)$ is also a skew brace, and has given applications of this concept to the equivalent theory of Hopf-Galois structures. The goal of this paper is to deal with bi-skew braces $(G, \cdot, \circ)$ from the yet equivalent point of view of regular subgroups of the holomorph of $(G, \cdot)$. In particular, we find that certain groups studied by T. Kohl, F. Dalla Volta and the author, and C. Tsang all yield examples of bi-skew braces. Building on a construction of Childs, we also give various methods for …
Studying solutions of the Yang-Baxter equation through skew braces, with an application to indecomposable involutive solutions with abelian permutation group
Using Bieberbach groups we study multipermutation involutive solutions to the Yang-Baxter equation. We use a linear representation of the structure group of an involutive solution to study the unique product property in such groups. An algorithm to find subgroups of a Bieberbach group isomorphic to the Promislow subgroup is introduced and then used in the case of structure group of involutive solutions. To extend the results related to retractability to non-involutive solutions, following the ideas of Meng, Ballester-Bolinches and Romero, we develop the theory of right $p$-nilpotent skew brace…
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 braces can be used to construct solutions of the quantum Yang-Baxter equation. In this article, we introduce a notion of action of a skew brace, and show how it leads to solutions of the closely associated reflection equation.
Braces were introduced by W. Rump in 2006 as an algebraic system related to the quantum Yang-Baxter equation. In 2017, L. Guarnieri and L. Vendramin defined for the same purposes a more general notion of a skew left brace. Recently, L. Guo, H. Lang, Y. Sheng [arXiv:2009.03492] gave a definition of what is a Rota-Baxter operator on a group. We connect these two notions as follows. It is shown that every Rota-Baxter group gives rise to a skew left brace. Moreover, every skew left brace can be injectively embedded into a Rota-Baxter group. When the additive group of a skew left brace is complete,…
Free Skew Braces and Free Solutions of the Yang--Baxter Equation
Vrije Universiteit Brussel（布鲁塞尔自由大学）；Université libre de Bruxelles（布鲁塞尔自由大学）；Università degli Studi di Napoli Federico II（那不勒斯费德里科二世大学）；Maynooth University（梅努斯大学）
This is the first part of a series of two articles. In this paper we enumerate and classify the left braces of size $p^2q$ where $p$ and $q$ are distinct prime numbers by the classification of regular subgroups of the holomorph of the abelian groups of the same order. We also provide the formulas that define the constructed braces.
We define the radical and weight of a skew left brace and provide some basic properties of these notions. In particular, we obtain a Wedderburn type decomposition for Artinian skew left braces. Furthermore, we prove analogues of a theorem of Wiegold, a theorem of Schur and its converse in the context of skew left braces. Finally, we apply these results to detect torsion in the structure group of a finite bijective non-degenerate set-theoretic solution of the Yang-Baxter equation.
Given a skew left brace $B$, a method is given to construct all the non-degenerate set-theoretic solutions $(X,r)$ of the Yang Baxter equation such that the associated permutation group $\mathcal{G}(X,r)$ is isomorphic, as a skew left brace, to $B$. This method depends entirely on the brace structure of $B$. We then adapt this method to show how to construct solutions with additional properties, like square-free, involutive or irretractable solutions. Using this result, it is even possible to recover racks from their permutation group.
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 involutive solutions. Roughly speaking, skew braces provide group-theoretical and ring-theoretical methods to understand solutions of the YBE. It turns out that skew braces appear in many different contexts, such as near-rings, matched pairs of groups, triply factorized groups, bijective 1-cocycles and Hopf-Galois extensions. These connections and some of their…
We classify all skew braces of Heisenberg type for a prime number $ p>3 $. Furthermore, we determine the automorphism group of each one of these skew braces (as well as their socle and annihilator). Hence, by utilising a link between skew braces and Hopf-Galois theory, we can determine all Hopf-Galois structures of Heisenberg type on Galois field extensions of fields of degree $ p^{3} $.
