Left Braces Research Articles

From actions of an abelian group on itself to left braces

Abstract We present a construction of left braces of right nilpotency class at most two based on suitable actions of an abelian group on itself with an invariance condition. This construction allows us to recover the construction of a free right nilpotent one-generated left brace of class two.

A note on right-nil and strong-nil skew braces

Nilpotency concepts for skew braces are among the main tools with which we are nowadays classifying certain special solutions of the Yang–Baxter equation, a consistency equation that plays a relevant role in quantum statistical mechanics and in many areas of mathematics. In this context, two relevant questions have been raised in F. Cedó, A. Smoktunowicz and L. Vendramin (Skew left braces of nilpotent type. Proc. Lond. Math. Soc. (3) 118 (2019), 1367–1392) (see questions 2.34 and 2.35) concerning right- and central nilpotency. The aim of this short note is to give a negative answer to both questions: thus, we show that a finite strong-nil brace B need not be right-nilpotent. On a positive note, we show that there is one (and only one, by our examples) special case of the previous questions that actually holds. In fact, we show that if B is a skew brace of nilpotent type and $b\ \ast \ b=0$ for all $b\in B$ , then B is centrally nilpotent.

Cohomology and extensions of relative Rota–Baxter groups

Relative Rota–Baxter groups are generalisations of Rota–Baxter groups and recently shown to be intimately related to skew left braces, which are well-known to yield bijective non-degenerate solutions to the Yang–Baxter equation. In this paper, we develop an extension theory of relative Rota–Baxter groups and introduce their low dimensional cohomology groups, which are distinct from the ones known in the context of Rota–Baxter operators on Lie groups. We establish an explicit bijection between the set of equivalence classes of extensions of relative Rota–Baxter groups and their second cohomology. Further, we delve into the connections between this cohomology and the cohomology of associated skew left braces. We prove that for bijective relative Rota–Baxter groups, the two cohomologies are isomorphic in dimension two.

Some properties of relative Rota–Baxter operators on groups

We find connection between relative Rota–Baxter operators and usual Rota–Baxter operators. We prove that any relative Rota–Baxter operator on a group H with respect to ( G , Ψ ) defines a Rota–Baxter operator on the semi-direct product H ⋊ Ψ G . On the other side, we give condition under which a Rota–Baxter operator on the semi-direct product H ⋊ Ψ G defines a relative Rota–Baxter operator on H with respect to ( G , Ψ ) . We introduce homomorphic post-groups and find their connection with λ-homomorphic skew left braces. Further, we construct post-group on arbitrary group and a family of post-groups which depends on integer parameter on any two-step nilpotent group. We find all verbal solutions of the quantum Yang-Baxter equation on two-step nilpotent group.

On the structure of left braces satisfying the minimal condition for subbraces

We analyse the structure of infinite weakly soluble left braces that satisfy the minimal condition for subbraces. We observe that they can be characterised as the left braces with Chernikov additive group. We also present an example of left braces satisfying the minimal condition for ideals, but that do not satisfy the minimal condition for subbraces.

Relative Rota–Baxter groups and skew left braces

Abstract Relative Rota–Baxter groups are generalizations of Rota–Baxter groups and have been introduced recently in the context of Lie groups. In this paper, we explore connections of relative Rota–Baxter groups with skew left braces, which are well known to give bijective non-degenerate set-theoretical solutions of the Yang–Baxter equation. We prove that every relative Rota–Baxter group gives rise to a skew left brace, and conversely, every skew left brace arises from a relative Rota–Baxter group. It turns out that there is an isomorphism between the two categories under some mild restrictions. We propose an efficient GAP algorithm, which would enable the computation of relative Rota–Baxter operators on finite groups. In the end, we introduce the notion of isoclinism of relative Rota–Baxter groups and prove that an isoclinism of these objects induces an isoclinism of corresponding skew left braces.

Schur multiplier and Schur covers of relative Rota–Baxter groups

Relative Rota–Baxter groups are generalizations of Rota–Baxter groups and share a close connection with skew left braces. These structures are well-known for offering bijective non-degenerate set-theoretical solutions to the Yang–Baxter equation. This paper builds upon the recently introduced extension theory and low-dimensional cohomology of relative Rota–Baxter groups. We prove an analogue of the Hochschild–Serre exact sequence for central extensions of relative Rota–Baxter groups. We introduce the Schur multiplier MRRB(A) of a relative Rota–Baxter group A=(A,B,β,T), and prove that the exponent of MRRB(A) divides |A||B| when both A and B are finite. We define weak isoclinism of relative Rota–Baxter groups, introduce their Schur covers, and prove that any two Schur covers of a finite bijective relative Rota–Baxter group are weakly isoclinic. The results align with recent results of Letourmy and Vendramin for skew left braces.

Split epimorphisms and Baer sums of skew left braces

We investigate the split epimorphisms in the categories of digroups and skew left braces. We show that, unlike the category DiGp of digroups, the category SkB of skew left braces is strongly protomodular. From that, we describe the expected Baer sums of exact sequences of skew left braces with abelian kernel.

A characterization of finite simple set-theoretic solutions of the Yang-Baxter equation

In this paper we present a characterization of finite simple involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation by means of left braces and we provide some significant examples.

Skew left braces and 2-reductive solutions of the Yang-Baxter equation

We study 2-reductive non-involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation. We give a combinatorial construction of any such solution of any (even infinite) size. We also prove that solutions associated to a skew left brace are 2-reductive if and only if the skew left brace is nilpotent of class 2. Moreover, all such skew left braces are actually bi-skew left braces. We focus on these structures and we give several equivalent properties characterizing solutions associated to bi-skew left braces.

Enumeration of left braces with additive group 𝐶₄×𝐶₄×𝐶₄

We show that the number of isomorphism classes of left braces of order 64 64 with additive group isomorphic to C 4 × C 4 × C 4 C_4\times C_4\times C_4 is 1 515 429 1\,515\,429 .

On the structure of some nilpotent braces

We prove a criteria for nilpotency of left braces in terms of the $\star$-central series and also discuss Noetherian braces, obtaining some of their elementary properties. We also show that if a finitely generated brace $A$ is Smoktunowicz-nilpotent, then the additive and multiplicative groups of $A$ are likewise finitely generated.

Some group-theoretical approaches to skew left braces

Some group-theoretical approaches to skew left braces

A Jordan–Hölder theorem for skew left braces and their applications to multipermutation solutions of the Yang–Baxter equation

Skew left braces arise naturally from the study of non-degenerate set-theoretic solutions of the Yang–Baxter equation. To understand the algebraic structure of skew left braces, a study of the decomposition into minimal substructures is relevant. We introduce chief series and prove a strengthened form of the Jordan–Hölder theorem for finite skew left braces. A characterization of right nilpotency and an application to multipermutation solutions are also given.

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Inducing braces and Hopf Galois structures

Let p be a prime number and let n be an integer not divisible by p and such that every group of order np has a normal subgroup of order p. (This holds in particular for p>n.) Under these hypotheses, we obtain a one-to-one correspondence between the isomorphism classes of braces of size np and the set of pairs (Bn,[τ]), where Bn runs over the isomorphism classes of braces of size n and [τ] runs over the classes of group morphisms from the multiplicative group of Bn to Zp⁎ under a certain equivalence relation. This correspondence gives the classification of braces of size np from the one of braces of size n. From this result we derive a formula giving the number of Hopf Galois structures of abelian type Zp×E on a Galois extension of degree np in terms of the number of Hopf Galois structures of abelian type E on a Galois extension of degree n. For a prime number p≥7, we apply the obtained results to describe all left braces of size 12p and determine the number of Hopf Galois structures of abelian type on a Galois extension of degree 12p.

Symmetric skew braces and brace systems

Abstract For a skew left brace ( G , ⋅ , ∘ ) {(G,\cdot\,,\circ)} , the map λ : ( G , ∘ ) → Aut ⁡ ( G , ⋅ ) {\lambda:(G,\circ)\to\operatorname{Aut}(G,\cdot\,)} , a ↦ λ a , {a\mapsto\lambda_{a},} where λ a ⁢ ( b ) = a - 1 ⋅ ( a ∘ b ) {\lambda_{a}(b)=a^{-1}\cdot(a\circ b)} for all a , b ∈ G {a,b\in G} , is a group homomorphism. Then λ can also be viewed as a map from ( G , ⋅ ) {(G,\cdot\,)} to Aut ⁡ ( G , ⋅ ) {\operatorname{Aut}(G,\cdot\,)} , which, in general, may not be a homomorphism. A skew left brace will be called λ-anti-homomorphic (λ-homomorphic) if λ : ( G , ⋅ ) → Aut ⁡ ( G , ⋅ ) {\lambda:(G,\cdot\,)\to\operatorname{Aut}(G,\cdot\,)} is an anti-homomorphism (a homomorphism). We mainly study such skew left braces. We device a method for constructing a class of binary operations on a given set so that the set with any two such operations constitutes a λ-homomorphic symmetric skew brace. Most of the constructions of symmetric skew braces dealt with in the literature fall in the framework of our construction. We then carry out various such constructions on specific infinite groups.

On ρ-conjugate Hopf–Galois structures

AbstractThe Hopf–Galois structures admitted by a Galois extension of fields $ L/K $ with Galois group G correspond bijectively with certain subgroups of $ \mathrm{Perm}(G) $. We use a natural partition of the set of such subgroups to obtain a method for partitioning the set of corresponding Hopf–Galois structures, which we term ρ-conjugation. We study properties of this construction, with particular emphasis on the Hopf–Galois analogue of the Galois correspondence, the connection with skew left braces, and applications to questions of integral module structure in extensions of local or global fields. In particular, we show that the number of distinct ρ-conjugates of a given Hopf–Galois structure is determined by the corresponding skew left brace, and that if $ H, H^{\prime} $ are Hopf algebras giving ρ-conjugate Hopf–Galois structures on a Galois extension of local or global fields $ L/K $ then an ambiguous ideal $ \mathfrak{B} $ of L is free over its associated order in H if and only if it is free over its associated order in Hʹ. We exhibit a variety of examples arising from interactions with existing constructions in the literature.

Left braces of size 8p

We describe all left braces of size 8p for an odd prime p≠3,7 and validate the number given by Bardakov, Neschadim and Yadav in [2]. We give a characterization for isomorphism classes of a semidirect product of left braces and then the description is done by first describing left braces of size 8, as conjugacy classes of regular subgroups of the corresponding holomorph, and then checking how many non isomorphic left braces of size 8p are obtained from each one of them.

Extensions and Well’s type exact sequence of skew braces

In this paper, we describe the split exact sequences of skew left braces. We define a free action of the second cohomology group of a skew left brace [Formula: see text] by [Formula: see text] on [Formula: see text] and show that this action becomes transitive if [Formula: see text] is a trivial skew brace. We also generalize the Well’s type exact sequence for extensions by the trivial skew brace.