1 Introduction↩︎

The Yang–Baxter equation has its origins in the works of Yang [1] and Baxter [2]. It has significant implications not only in mathematical physics but also in various branches of pure mathematics, including quantum groups, Hopf algebras, and knot theory [3]–[7]. A comprehensive description of all solutions of the Yang–Baxter equation remains currently beyond reach. An interesting family of solutions consists of those where the solution acts on a basis. Drinfeld referred to such solutions as set-theoretic in [8]. A set-theoretic solution of the Yang–Baxter equation is a pair \((X,r)\), where \(X\) is a non-empty set and \(r\colon X\times X\to X\times X\) is a map satisfying the braid equation \((r\times{\mathop{\mathrm{id}}})({\mathop{\mathrm{id}}}\times r)(r\times{\mathop{\mathrm{id}}})=({\mathop{\mathrm{id}}}\times r)(r\times {\mathop{\mathrm{id}}})({\mathop{\mathrm{id}}}\times r)\) on \(X\times X\times X\). The class of non-degenerate set-theoretic solutions holds particular importance. As usual we write \[r(x,y)=(\lambda_x(y),\rho_y(x))\] and thus non-degeneracy means that each \(\lambda_x\) and \(\rho_y\) is a bijection. Recall that by [9] non-degeneraracy implies that \(r\) is necessarily a bijective map provided \(X\) is finite. Recently, Jedlička and Pilitowska [10] have shown that any non-degenerate solution is bijective. Hence, for simplicity, we refer to non-degenerate bijective solutions simply as non-degenerate solutions. Pioneer works [11], [12] brought algebraic tools to study a subclass of these solutions, the non-degenerate set-theoretic solutions that are involutive, i.e., also \(r^2=\mathop{\mathrm{id}}\). Later in [13], [14] these methods were extended to the class of all non-degenerate solutions. Given a non-degenerate set-theoretic solution of the Yang–Baxter \((X,r)\), the (left) derived solution of \((X,r)\), for which we will use the suggestive notation \((X,r')\), is given by \[r'(x,y)=(y,\sigma_y(x)),\quad\text{where}\quad\sigma_y(x)=\lambda_y\rho_{\lambda_x^{-1}(y)}(x).\] It is well-known that \((X,r')\) also is a non-degenerate set-theoretic solution, see for instance [15]. It turns out that the derived solution plays a crucial role in the study of non-degenerate solutions, for example, being involutive or being injective for a non-degenerate solution can be read from its derived solution [14], [16]. It is easy to verify that \(X\) with respect to the binary operation \(\triangleleft\) defined by \(x\triangleleft y=\sigma_y(x)\) is a rack, i.e., \(\sigma_y\) is bijective for all \(y\in X\) and for any \(x,y,z\in X\) it holds \((x\triangleleft y)\triangleleft z=(x\triangleleft z)\triangleleft (y\triangleleft z)\). A rack \((X,\triangleleft)\) is said to be a quandle if \(x \triangleleft x=x\) for each \(x\in X\). Any group \(B\) with the operation given by \(g\triangleleft h=h^{-1}gh\) defines a rack operation on \(B\), which is clearly a quandle operation. Every subrack of such a quandle is called conjugation quandle. On the other hand, a rack \((X,\triangleleft)\) defines a set-theoretic solution of the Yang–Baxter equation by \(r(x,y)=(y,x\triangleleft y)\). This relation between solutions of the Yang–Baxter equation and racks has been noticed by Brieskorn in [17]. The notion of racks has been considered in the literature, mainly as a way to produce knot invariants [7], [17]–[19] and in the study of Nichols algebras over a group [20].

A characterisation of arbitrary finite non-degenerate solutions is presently beyond reach. Hence one needs to restrict the investigations to building blocks of such solutions. Of course a first and obvious restriction is to deal with indecomposable solutions. Recall that a solution \((X,r)\) is said to be decomposable if \(X=Y\cup Z\), a disjoint union of non-empty subsets, and \(r\) restricted to \(Y\times Y\) and \(Z\times Z\) respectively defines a solution on \(Y\) and \(Z\).

Another obvious restriction is to simple solutions, i.e., a solution \((X,r)\) of which any epimorphic image is either isomorphic to \((X,r)\) or to a solution \((Y,s)\) with \(Y\) a singleton. Clearly any finite non-degenerate solution has an epimorphic image that is simple.

We note that a non-degenerate solution \((X,r)\) with \(X\) of cardinality \(2\) is simple. Moreover, if the cardinality of \(X\) is at least \(3\) and \((X,r)\) is simple then the solution \((X,r)\) is also indecomposable. Simple solutions are the smallest non-trivial solutions and thus are the building blocks of all solutions. Note that the only simple decomposable solution \((X,r)\) is the flip map on two elements, i.e., \(r(x,y)=(y,x)\), therefore throughout the paper we always assume that \(|X|\geqslant 3\) and thus a simple solution is indecomposable. The definition of simple non-degenerate derived solutions appeared in [21], where a group-theoretic characterisation of simple quandles has been obtained. The definition of simple solution seems to have appeared first in [20], where the classification of simple racks has been improved given a concrete characterisation of which groups give rise to a simple rack. Concerning involutive non-degenerate solutions, in [22], Vendramin introduced a notion of simple involutive non-degenerate solutions (using the language of cycle sets). This definition of simplicity, however, coincides with the original one only within the class of finite indecomposable non-degenerate involutive solutions. Cedó and Okniński [23], [24] (see also [25]) constructed finite simple non-degenerate involutive solutions. Recently in [26] they proved for a finite simple non-degenerate involutive solution \((X,r)\) that if \(|X|\) is not a prime power then \(|X|\) cannot be square-free. Moreover, if \(|X|\) is a prime power \(p^n\) then \(n=1\), and there exist simple solutions of order \(p\) and there is only one involutive up to isomorphism. In [21] Joyce described simplicity for quandles in a group theoretic setting and Castelli, in [27], characterised the simplicity of non-degenerate involutive solutions in the algebraic setting of skew left braces.

Recall that Rump [28] and Guarnieri and Vendramin [29] introduced skew left braces to provide an algebraic framework for studying arbitrary non-degenerate set-theoretic solutions of the Yang–Baxter equation. A skew left brace is a triple \((B,+,\circ)\) such that both \((B,+)\) and \((B,\circ)\) are groups and \[a\circ(b+c)=a\circ b-a+a\circ c\] holds for all \(a,b,c\in B\). The theory of Hopf–Galois extensions suggests to use the following terminology based on properties of the additive group. One says that a skew left brace \(B\) is of abelian type if \((B,+)\) is an abelian group (Rump in [28] simply names it a left brace) and of cyclic type if \((B,+)\) is a cyclic group. Since their introduction, skew left braces have been widely studied [16], [25], [30], [31], and recently are popularised in [32]. Every skew left brace \(B\) naturally provides a set-theoretic solution \((B,r_B)\) of the Yang–Baxter equation that is non-degenerate and skew left braces of abelian type provide involutive solutions. Specifically, \(r_B\colon B\times B\to B\times B\) is given by \[r_B(a,b)=(-a+a\circ b, (-a+a\circ b)^{-1}\circ a\circ b),\] where \(c^{-1}\) denotes the inverse of \(c\in B\) in the group \((B,\circ)\). Two fundamental skew left braces are associated with a finite non-degenerate solution \((X,r)\): the structure skew left brace \(G(X,r)\) and the permutation skew left brace \(\mathcal{G}(X,r)\). In the next section we recall rigorous definitions of these structures. Let us just mention that if the solution \((X,r)\) is finite so is the permutation skew left brace. Moreover, if two non-degenerate solutions \((X,r)\) and \((Y,s)\) are isomorphic, then we have \(G(X,r)\cong G(Y,s)\) as skew left braces, as well as the permutation groups \(\mathcal{G}(X,r)\cong \mathcal{G}(Y,s)\). But the converse is not true. Given a finite skew left brace \(B\), Bachiller, Cedó, and Jespers [33], [34] provided a construction of all finite non-degenerate set-theoretic solutions \((X,r)\) such that \(\mathcal{G}(X,r)\) and \(B\) are isomorphic as skew left braces. Providing more evidence that skew left braces are the right algebraic tool to investigate all non-degenerate solutions of the Yang–Baxter equation. Within this framework, there is the need to investigate its building blocks, the simple skew left braces, i.e., skew left braces with no non-trivial homomorphic images. Note that in contrast with finite simple groups, it has been shown [24] that there is an abundance of finite simple skew left braces of abelian type [24], [35]. Moreover, more simple skew left braces of abelian type have been constructed in [36]. Only recently, Byott [37] constructed a family of finite simple skew left braces that are not of abelian type. The first instances of this family can be found in the GAPpackage YangBaxter [38], i.e., SmallSkewbrace(12,22) and SmallSkewbrace(12,23).

Okniński and Cedó [23] proved that some finite simple skew left braces of abelian type determine an involutive simple solution \((X,r)\). They showed that finite simple braces which are additively generated by an orbit \(X\) under the action of the permutation skew left brace \(\mathcal{G}(X,r)\) yield simple non-degenerate involutive solutions on \(X\). This connection between involutive simple solutions and simple skew left braces of abelian type recently has been refined by Castelli in [27] by showing that the simplicity of a finite non-degenerate involutive solution \((X,r)\) is described by the algebraic structure of its permutation skew left brace.

Problem 10, posed by Vendramin in [39], asks whether the simplicity of an involutive solution \((X,r)\) can be read in terms of the left brace \(G(X,r)\) and expresses interest in developing the theory of non-involutive simple solutions, particularly in understanding how simplicity can be reflected in the skew left brace where the solution is realised.

In this paper, we address this problem by developing a new algebraic tool to describe finite simple non-degenerate solutions to the Yang—Baxter equation. Our approach provides a comprehensive solution, offering a framework that resolves the issue of reflecting simplicity in skew left braces and extends existing results. While similar work has been done for specific subclasses of solutions, such as quandles and involutive solutions, through the works of Joyce [7] and Castelli [27], our method not only recovers these approaches but also advances them. We also apply our method to construct new infinite families of simple solutions.

The first step is to identify two quite distinct natural families of simple non-degenerate finite solutions:

• Lyubashenko type solutions, which are fully classified using combinatorial methods, see Section 4, and

• solutions that arise as subsolutions of those associated with skew left braces that have a clear algebraic structure, see part (2) in the following theorem.

This can be summarised as follows: we use the natural action of a skew left brace \(B\) on itself, i.e., the action with respect to the group generated by the maps \(\lambda_a\) and \(\sigma_a\) with \(a\in B\). Such an action can be restricted to ideals and subsets of \(B\) that define subsolutions. The formulation also makes use of the ideal \(B^{(2)}=B*B\), i.e., the additive subgroup of \(B\) generated by all the elements \(a*b=-a+a\circ b-b\) for \(a,b\in B\), and the ideal \(B^{(3)}=B^{(2)}*B\), that is the additive subgroup of \(B\) generated by all elements of the form \(x*y\) with \(x\in B^{(2)}\) and \(y\in B\).

Recall that a non-degenerate solution \((X,r)\) is said to be Lyubashenko (see [8]) if \(r(x,y)=(\lambda(y),\rho(x))\) for all \(x,y\in X\), where \(\lambda\) and \(\rho\) are commuting permutations on \(X\). Solutions of type (1) are completely characterised in Section 4, namely we prove that the cardinality of \(X\) is a prime number \(p\) and the group generated by \(\lambda\) and \(\rho\) is cyclic of order \(p\).

As for the solutions of type (2) we can naturally identify three subcases:

1. \(B^{(2)}=0\) and thus \(V=[B,B]\),

2. \(B^{(2)}\ne 0\) and \(B^{(3)}=0\) and thus \(V=B^{(2)}\),

3. \(B^{(2)}=B^{(3)}\ne 0\).

Let us examine these subcases. If \(B^{(2)}=0\) this means that the additive and multiplicative structures of the skew braces coincide and the associated solution is a derived solution. Hence, this reduces to the study of simple quandles, yielding the same classification obtained by Joyce in [21] and Andruskiewitsch and Graña in [20].

We achieve a complete classification of case (2) in Section 5. To do so, we need to split the classification into two cases based on the abelianity of \(V\).

• If \(V\) is abelian we obtain the characterisation in Theorem 17. In summary, we prove that \(V\) must be an elementary abelian \(p\)-group and that the additive structure of \(B\) is given by the semi-direct product of \(V\) and a cyclic group of order \(k\). We also provide a concrete expression for the lambda map of \(B\), which allows us to derive an explicit formula for the simple solution obtained from this skew brace.

• If \(V\) is non-abelian we obtain the characterisation in Theorem 20. Here we prove that \(V\) must be a finite non-abelian characteristically simple group and the additive group of \(B\) is the quotient of a semi-direct product of \(V\) with a cyclic group over the center. In this case as well, we provide an explicit formula for the simple solution obtained from this skew brace.

To summarise we have complete control over the case when \(B^{(3)}=0\). It is also worth mentioning that all solutions obtained within this classification are neither involutive nor derived (since we are assuming \(B^{(2)}\ne 0\)). Hence, they provide an abundance of new simple solutions complementary to all the work done in the context of involutive simple solutions (see [23]–[26]).

Case (3) is open, including simple solutions that have the associated skew brace simple and, more generally, perfect. As one might expect, we do not have a complete classification. However, our theorem can be applied to existing families of simple skew braces to obtain families of solutions, as shown in Example 29, where we construct an infinite family of simple solutions based on the family of simple skew braces constructed by Byott in [37].

2 Preliminaries↩︎

Let \((X,r)\) be a finite non-degenerate solution and \((X,r')\) its (left) derived solution. We denote by \[M=M(X,r)=\langle X\mid x\circ y=\lambda_x(y)\circ\rho_y(x)\text{ for all }x,y\in X \rangle\] and \[A=A(X,r)=\langle X\mid x+y=y+\sigma_y(x)\text{ for all }x,y\in X \rangle\] the Yang–Baxter monoid (also known as structure monoid of \((X,r)\)) and the left derived Yang–Baxter monoid of \((X,r)\) (also known as derived structure monoid of \((X,r)\)), respectively (note that \(A(X,r)=M(X,r')\)). Moreover, \[G=G(X,r)=\mathop{\mathrm{gr}}(X\mid x\circ y=\lambda_x(y)\circ\rho_y(x)\text{ for all }x,y\in X)\] denotes the Yang–Baxter group (also known as structure group) of \((X,r)\) and \[A_{\mathop{\mathrm{gr}}}=A_{\mathop{\mathrm{gr}}}(X,r)=\mathop{\mathrm{gr}}(X\mid x+y=y+\sigma_y(x)\text{ for all }x,y\in X)\] denotes the left derived Yang–Baxter group (also known as left derived structure group) of \((X,r)\) (here \(\langle X\mid S \rangle\) and \(\mathop{\mathrm{gr}}(X\mid S)\) refer, respectively, to a monoid and group presentation with generators from a set \(X\) and relations from a set \(S\)).

It was shown by Gateva-Ivanova and Majid [40] that the maps \(\lambda\colon X\to\mathop{\mathrm{Sym}}(X)\colon x\mapsto\lambda_x\) and \(\rho\colon X\to\mathop{\mathrm{Sym}}(X)\colon x\mapsto \rho_x\) can be extended uniquely to \(M=M(X,r)\) (for simplicity we use the same notation for the extension) so that one obtains a monoid morphism \(\lambda\colon(M,\circ)\to\mathop{\mathrm{Sym}}(M)\colon a\mapsto\lambda_a\), and a monoid anti-morphism \(\rho\colon(M,\circ)\to\mathop{\mathrm{Sym}}(M)\colon a\mapsto\rho_a\). Moreover, the map \(r_M\colon M\times M\to M\times M\), defined as \(r_M(a,b)=(\lambda_a(b),\rho_b(a))\), is a non-degenerate solution of the Yang–Baxter equation. Furthermore, for \(a,b,c\in M\), we have: \[\begin{align} a\circ b &=\lambda_a(b)\circ\rho_b(a),\tag{1}\\ \lambda_a(b\circ c) &=\lambda_a(b)\circ\lambda_{\rho_b(a)}(c),\tag{2}\\ \rho_a(b\circ c) &=\rho_{\lambda_c(a)}(b)\circ\rho_a(c).\tag{3} \end{align}\] Also the map \(\sigma\colon X\to\mathop{\mathrm{Sym}}(X)\colon x\mapsto \sigma_x\) can be extended to a monoid anti-morphism \(\sigma\colon A\to\mathop{\mathrm{Sym}}(M)\) given by \(\sigma_b(a)=\lambda_b\rho_{\lambda^{-1}_a(b)}(a)\). One obtains in the monoid \((A,+)\) that \(a+b=b+\sigma_b(a)\) for all \(a,b\in A\). It follows (see, e.g., [41]) that there exists a bijective \(1\)-cocycle \(\pi\colon M\to A\) (via the \(\lambda\)-map, i.e., \(a+\lambda_a(b)=a\circ b\) for \(a,b\in A\)) such that \(\pi(x)=x\) for all \(x\in X\). So, we have a monoid embedding \[f\colon M\to A\rtimes{\mathop{\mathrm{Im}}(\lambda)}\colon m\mapsto(\pi(m),\lambda_m).\] Abusing notation, we will often identify \(m\) with \(f(m)\) or \(\pi(m)\). Hence, in particular, we may also denote by \(x\), with \(x\in X\), the generators of \(A\). So, we simply may write \[M=\{a=(a,\lambda_a):a\in A\}=\langle x=(x,\lambda_x):x\in X \rangle.\] Hence, in the terminology of [9], \((M,+,\circ,\lambda,\sigma)\) is a unital YB-semitruss, that is \((M,+)\) and \((M,\circ)\) are monoids (with the same identity \(0=1\)), \(\lambda\colon (M,\circ) \to \mathop{\mathrm{Aut}}(M,+)\) is a monoid morphism with \[a\circ(b+c)=a\circ b+\lambda_a (c)\quad\text{and}\quad a+\lambda_a(b)=a\circ b,\] and thus \[\lambda_a\lambda_b=\lambda_{\lambda_a(b)}\lambda_{\rho_b(a)},\label{eq:YBE1}\tag{4}\] \(\sigma\colon (M,+)\to\mathop{\mathrm{Aut}}(M,+)\) is a monoid anti-morphism with \[a+b=b+\sigma_b(a)\quad\text{and}\quad\sigma_{\lambda_a(b)}\lambda_a(c)=\lambda_a\sigma_b(c),\label{eq:lamsig}\tag{5}\] and thus also \[\sigma_a\sigma_b=\sigma_{\sigma_a(b)}\sigma_a\label{eq:sigsig}\tag{6}\] for all \(a,b,c\in M\).

Because of the latter assumption, one obtains that the maps \(\lambda\), \(\rho\), and \(\sigma\) naturally induce left and right actions on the structure group (see also [42]) and so that the equations 1 , 2 , and 3 remain valid in \(G\); we use the same notation for these maps. Moreover, the bijective \(1\)-cocycle \(M\to A\) naturally induces a bijective \(1\)-cocycle \(G\to A_{\mathop{\mathrm{gr}}}\) (via the \(\lambda\)-map). Hence again we can identify \(G\) with \(A_{\mathop{\mathrm{gr}}}\) and thus \((G,+,\circ,\lambda,\sigma)\) is a YB-semitruss with \((G,+)\) and \((G,\circ)\) groups, in other words it is a skew left brace. Also, \[G=\{(a,\lambda_a):a\in A_{\mathop{\mathrm{gr}}}\} \subseteq A_{\mathop{\mathrm{gr}}}\rtimes {\mathop{\mathrm{Im}}(\lambda)}.\] So, here we thus have \[\lambda_a (b)=-a+a\circ b\quad\text{and}\quad\sigma_b(a)=-b+a+b\] for all \(a,b\in G\), and we have the associated solution \((G,r_G)\) with \(r_G(a,b)=(\lambda_a (b),\rho_b(a))\). Note that \(X\) is naturally embedded in the structure monoid \(M\). However, the natural map \(i\colon X\to G\) is not necessarily injective (see, e.g., [16]) but it is a morphism of solutions. A solution \((X,r)\) is said to be injective if \(i\) is injective. It is well known that involutive solutions are always injective.

In [43] the group, called the permutation group of the non-degenerate solution \((X,r)\), \[\mathcal{G}_{\lambda,\rho}=\mathcal{G}_{\lambda,\rho}(X,r)=\mathop{\mathrm{gr}}((\lambda_x,\rho_x^{-1}):x\in X)\subseteq\mathop{\mathrm{Sym}}(X)^2,\] has been introduced and it is shown that \(\mathcal{G}_{\lambda,\rho}\) is equipped with a skew left brace structure so that the natural map \(G\to\mathcal{G}_{\lambda,\rho}\) is an epimorphism of skew left braces. We will consider an isomorphic copy of this skew left brace. For this we first need to prove some lemmas. The following lemma has been shown partially in [9]. However an identity shown in [9] gives an easier proof and a complete symmetric result. Put \[\mathfrak{q}\colon X\to X\colon x\mapsto\lambda_x^{-1}(x),\] known as the diagonal map. Recall that \(\mathfrak{q}\) is a bijective map because \((X,r)\) is non-degenerate and thus bijective (see [9], [44] and [10]).

As a consequence of the previous lemma we can now extend [9] in the following sense. For completeness’ sake we include a proof.

We now introduce the set \[\mathcal{G}_{\sigma,\lambda}=\mathcal{G}_{\sigma,\lambda}(X,r)=\{\mathop{\mathrm{v}}_a=(\sigma_a^{-1},\lambda_a):a\in A\}.\] Because of Lemma 2 we have the following well-defined binary operation on this set: \[\mathop{\mathrm{v}}_a+\mathop{\mathrm{v}}_b=\mathop{\mathrm{v}}_{a+b}\quad\text{and}\quad\mathop{\mathrm{v}}_a\circ\mathop{\mathrm{v}}_b=\mathop{\mathrm{v}}_{a\circ b}.\] Clearly \(\mathcal{G}_{\sigma, \lambda}\) is additively (and multiplicatively) generated by the set \(\{\mathop{\mathrm{v}}_x:x\in X\}\). As in [9] one now obtains that \(\mathcal{G}_{\sigma ,\lambda}\) is a skew left brace for these operations with \(\lambda\)-map \[\lambda_{\mathop{\mathrm{v}}_a}(\mathop{\mathrm{v}}_b)=\mathop{\mathrm{v}}_{\lambda_a(b)}\] and \(\sigma\)-map \[\sigma_{\mathop{\mathrm{v}}_a}(\mathop{\mathrm{v}}_b)=\mathop{\mathrm{v}}_{\sigma_a(b)}.\] (Note that we use a \(\lambda\) notation in several contexts, for example for defining set-theoretical solutions and on the skew left brace \(\mathcal{G}_{\sigma,\lambda}\). It is clear from the context and the index used, which \(\lambda\)-map we consider. The same convention applies to the \(\sigma\)-map.)

Since the image of the map \[\mathop{\mathrm{v}}\colon M=A\to\mathcal{G}_{\sigma,\lambda}\colon a\mapsto{\mathop{\mathrm{v}}_a}\] consists of invertible elements, both additively and multiplicatively, it induces an epimorphism of skew left braces (for which we use the same notation) \[\mathop{\mathrm{v}}\colon G\to\mathcal{G}_{\sigma,\lambda}\colon a\mapsto{\mathop{\mathrm{v}}_a}.\] Note that it also follows from Lemma 1 that the map \(\mathcal{G}_{\sigma,\lambda}\to\mathcal{G}_{\lambda,\rho}\), defined as \({\mathop{\mathrm{v}}_a}\mapsto(\lambda_a,\rho_a^{-1})\), is an isomorphism of skew left braces. From now on we work with the skew left brace \(\mathcal{G}_{\sigma,\lambda}\) and we simply denote it as \[\mathcal{G}=\mathcal{G}_{\sigma,\lambda}.\] Recall that the socle of the skew left brace \(G\) is by definition \[\mathop{\mathrm{Soc}}(G)=\mathop{\mathrm{Ker}}(\mathop{\mathrm{v}})=\mathop{\mathrm{Ker}}(\lambda)\cap Z(G,+)=\{a\in G:\lambda_a=\sigma_a=\mathop{\mathrm{id}}\}.\] It is an ideal of the skew left brace \(G\). Ideals of skew left braces are precisely the kernels of morphisms of skew left braces. It turns out that an additive subgroup \(I\) of a skew left brace \((B,+,\circ)\) is an ideal if and only if it is \(\lambda\)-invariant, i.e., \(\lambda_a (I)=I\) for all \(a\in B\), \((I,+)\) is a normal subgroup of \((B,+)\) and \((I,\circ)\) is a normal subgroup of \((B,\circ)\). Equivalently, \(I\) is \(\lambda\)-invariant, \((I,+)\) is a normal subgroup of \((B,+)\) and \(i*b\in I\) for all \(i\in I\) and \(b\in B\). Recall that the operation \(*\) is defined as follows: \[a*b=-a+a\circ b-b=\lambda_a(b)-b\] for all \(a,b\in B\). An additive subgroup \(I\) that is \(\lambda\)-invariant is said to be a left ideal of the skew left brace.

The solution \((X,r)\) is said to be irretractable [16] (see also, e.g., [12], [43]) if the restriction of the map \(\mathop{\mathrm{v}}\colon M\to\mathcal{G}_{\sigma,\lambda}\) to \(X\) is injective. In case \(\mathop{\mathrm{v}}(X)\) is a singleton then the solution \((X,r)\) is a Lyubashenko solution.

Clearly, the restriction of the map \(\mathop{\mathrm{v}}\) to \(X\) is a morphism of solutions. Hence, if \((X,r)\) is simple then it follows that either \(\mathop{\mathrm{v}}(X)\) is a singleton and we have a Lyubashenko solution, or the restriction of \(\mathop{\mathrm{v}}\) to \(X\) is an injective map, i.e., the solution \((X,r)\) is irretractable. Note that in the latter case \((X,r)\) is an injective solution and it is isomorphic with the solution induced on \(\mathop{\mathrm{v}}(X)\). Furthermore, in the latter case \(\mathcal{G}_{\sigma,\lambda}\) is additively (and also multiplicatively) generated by the set \(\mathop{\mathrm{v}}(X)\).

Let \[\mathcal{C}=\{\sigma_a:a\in A\}.\] It is a group for the multiplication \[\sigma_a\circ\sigma_b=\sigma_a\sigma_b=\sigma_{b+a}\] (note that the multiplication \(\circ\) in \(\mathcal{C}\) coincides, in fact, with the ordinary composition of maps) and we consider this as a trivial skew left brace, i.e., its additive structure is the same as its multiplicative structure (i.e., \(\sigma_a + \sigma_b = \sigma_a\circ\sigma_b\)). The multiplicative group \(\mathcal{G}\) acts on \(\mathcal{C}\) via the action \(\cdot\) defined as follows: \[{\mathop{\mathrm{v}}_a}\cdot{\sigma_b}=\sigma_{\lambda_a(b)}=\lambda_a\sigma_b\lambda_a^{-1}.\] Hence we may consider the semi-direct product of skew left braces \[\mathcal{S}=\mathcal{C}\rtimes \mathcal{G}.\] (For better readability we will denote elements of \(\mathcal{S}\) as pairs \((\sigma_a,\mathop{\mathrm{v}}_b)\) with \(a,b\in A\).) So, the additive operation is the componentwise addition and the multiplication is as follows: \[(\sigma_a,\mathop{\mathrm{v}}_b)\circ(\sigma_c,\mathop{\mathrm{v}}_d)=(\sigma_a\circ\sigma_{\lambda_b(c)},\mathop{\mathrm{v}}_{b\circ d})=(\sigma_{\lambda_b(c)+a},\mathop{\mathrm{v}}_{b\circ d}).\]

Note that the action of \(\mathcal{S}\) on \(X\) and the action of the group \(\langle \lambda_x,\rho_y\colon x,y\in X \rangle\) on \(X\) have the same orbits (see for example the proof of [45]).

3 Simple solutions versus skew left braces↩︎

In the skew left brace \(\mathcal{S}=\mathcal{C}\rtimes \mathcal{G}\), as is common, we identify \(\mathcal{C}\) and \(\mathcal{G}\) as sub skew left braces. In particular, we simply write \((\sigma_a,\mathop{\mathrm{v}}_0)\) as \(\sigma_a\) and \((\sigma_0,\mathop{\mathrm{v}}_a)\) as \(\mathop{\mathrm{v}}_a\).

We now formulate the main result in what appears at first sight a more general context. However the following reasoning shows this is actually not the case. Let \(B\) be a finite skew left brace and let \((B,r_B)\) be its associated solution. Hence we have the associated skew left brace \(\mathcal{S}(B)=\mathcal{S}(B,r_B)\) that acts on the set \(B\) via the group \((B,+)\rtimes (B,\circ)\). In other words, the element \((a,b)\), with \(a,b\in B\), acts on \(x\in B\) as follows \((a,b)\cdot x=\sigma_a\lambda_b(x)\). If \(X\) is a subset of \(B\) and \(I\) is an ideal of \(B\) then we say that \(I\) acts on \(X\) if \(X\) is invariant under the action of the subgroup \(I\rtimes I\) of \((B,+)\rtimes (B,\circ)\).

Assume \(X\) is a subset of \(B\) that is an additive set of generators of \((B,+)\). Assume furthermore that \(r_B\) restricts to a solution \(r=r_B|_{X\times X}\) on \(X\) (i.e., \(X\) is invariant under the action of \(\mathcal{S}(B)\)). Hence, we obtain natural morphisms of skew left braces \[G(X,r)\to G(B,r_B)\to B\to \mathcal{G}(B,r_B)\to \mathcal{G}(X,r)\] (the first morphism is induced by the natural embedding \((X,r)\to(B,r_B)\), the second follows because the morphism from the free group on \(B\) to \(B\) respects the defining relations of \(G(B,r_B)\), the third one is \(a\mapsto(\sigma^{-1}_a,\lambda_a)\), and the last one is the restriction). Since the composition of all these maps is the natural epimorphism \(\mathop{\mathrm{v}}\colon G(X,r)\to\mathcal{G}(X,r)\), the composed map \(B\to\mathcal{G}(B,r_B) \to \mathcal{G}(X,r)\) is an epimorphism. If also \((X,r)\) is not a Lyubashenko solution then there exist distinct \(x,y\in X\) with different images in \(\mathcal{G}(X,r)\). Hence if the ideal \(V\) generated by all \(x-y\), with \(x,y\in X\subseteq B\), is the smallest non-zero ideal of \(B\) then it follows that \(B\cong \mathcal{G}(B,r_B)\cong\mathcal{G}(X,r)\).

So, we obtain the following result.

If in the previous theorem \(B\) is a finite simple skew left brace then the conditions are equivalent with \(B\) is additively generated by \(X\) and the action of the group \(\langle \sigma_a\lambda_b:a,b\in B \rangle\) on \(X\) is transitive. It is worth mentioning that in [23] examples of finite simple skew left braces of abelian type \(B\) are given (so the additive structure is commutative) such that for every orbit \(\Omega\) of any element under the action of the \(\lambda\)-map, \(B\ne\langle \Omega \rangle_+\).

Assume \((B,+,\circ)\) and \(X\) are as in Theorem 6 and \(V\ne B\). Then \(B/V\) is a trivial skew left brace of cyclic type. In particular, \(B^{(2)}=B*B\subseteq V\). As \(B^{(2)}\) is an ideal of \(B\), and, by assumption \(V\) is the smallest non-zero ideal of \(B\), it follows that either \(B^{(2)}=0\) or \(B^{(2)}=V\). The former corresponds to the case where \(B\) is a trivial skew left brace, and thus \((X,r)\) is a left derived solution. In the latter case, \(B/B^{(2)}\) is a trivial skew left brace of cyclic type and \(B^{(2)}\) acts transitively on \(X\).

If \(B^{(2)}\ne 0\) then, since \(B^{(3)}=B^{(2)}*B\) is an ideal of \(B\), we get that either \(B^{(3)}=0\) or \(B^{(3)}=B^{(2)}\).

Assume, on the other hand, that \(B^{(2)}=0\). Note that then \(B\) cannot be of abelian type if \(|X|>2\) (otherwise, the solution is the twist map and it is simple if and only if \(|X|\leqslant 2\)). Since by Theorem 5 \((X,r)\) is irretractable, it also follows that \(Z(B)\) is trivial. Indeed, if \(a\in Z(B)\) and \(y\in X\), then \(\lambda_a=\mathop{\mathrm{id}}\) and \(\sigma_a(y)\in X\) and \(\sigma_a(y)=y\). So, \(\sigma_a=\mathop{\mathrm{id}}\) and \(\lambda_a=\mathop{\mathrm{id}}\), and since the map \(B\to \mathcal{G}(X,r)\), given as \(a\mapsto(\sigma_a^{-1},\lambda_a)\), is an isomorphism (see the comment just before Theorem 6), we obtain that \(a=0\). Moreover, the smallest non-zero ideal \(V\) needs to coincide with the derived subgroup \([B,B]\). This follows from the fact that \(X\) is an orbit and so all generating elements in \(V\) are commutators. Indeed, let \(x-y\) be a generator of \(V\), since \(X\) is an orbit there exists \(a\in B\) such that \(y=\sigma_a(x)=-a+x+a\), and thus \(x-y=x-a-x+a=[x,-a]\). Finally, note that if in this case also \([B,B]\ne B\) then, since \(X\) is a conjugacy class generating \(B\), the group \(B/[B,B]\) needs to be cyclic. Note that this recovers the classification of simple quandles given by Joyce in [21] and which is stated in part two of the following corollary. Hence we have shown Theorem 1 stated in the introduction.

We finish this section with a remark on the derived solution of a simple solution.

4 Lyubashenko solutions↩︎

In this section we describe the finite simple non-degenerate solutions that are of Lyubashenko type, that is solutions of type (1) in Theorem 1. In [47] Etingof, Guralnick, and Soloviev have described the indecomposable non-degenerate solutions of prime order. We now show that Lyubashenko solutions that are simple have to be of prime order and indecomposable.

Recall that a solution \((X,r)\) is said to be primitive if the group \(\mathcal{S}(X,r)\) acts primitively on \(X\) (cf. [45]). One may check that for a Lyubashenko solution this is the same as saying that \(\langle \lambda,\rho \rangle\) is a primitive permutation group.

Note that there are examples of non-involutive simple non-degenerate solutions of prime order that are not of Lyubashenko type. This is in contrast with the involutive case where there is only one involutive simple solution of prime order (see [12]) and it is of Lyubashenko type, i.e., of the form \(r(x,y)=(\lambda(y),\lambda^{-1}(x))\) with \(\lambda\) a full cycle on \(X\).

5 Solutions with \(B^{(3)}=0\)↩︎

In this section we shall focus on finite simple non-degenerate solutions \((X,r)\) which are of type (2) in Theorem 1. So, in particular, these are solutions that are embedded in a solution determined by a finite skew left brace \(B\) that has a unique minimal non-zero ideal \(V\) (recall that it necessarily contains all elements \(x-y\) with \(x,y\in X\)) such that \(B/V\) is a trivial skew left brace of cyclic type and \(B^{(3)}=0\).

Let us start with some elementary observations.

The condition \(B^{(3)}=0\) implies the the solution \((X,r)\) is of special type.

Note that the converse of Remark 12 is also true. Namely, if \((X,r)\) is a twisted rack solution, which is a subsolution of a solution associated to a skew left brace \(B\) with the listed properties, then \(B^{(3)}=0\). Indeed, in this case \(\lambda_a=\lambda^{|a|}\) for \(a\in B\), as \(B\) is generated by the set \(X\) (here \(|a|\) means the additive order of the image of \(a\) in \(B/V\)). In particular, \(\lambda_{y-x}=\mathop{\mathrm{id}}\) for any \(x,y\in X\), so \((y-x)*B=0\). Then, since \((a+b)*c=a*(\lambda_a^{-1}(b)*c)+\lambda_a^{-1}(b)*c+a*c\) for all \(a,b,c\in B\), it follows that \(V*B=0\) and thus \(B^{(3)}=B^{(2)}*B=0\).

The equality \(B^{(3)}=0\) implies also that \(V\) is a trivial skew left brace, i.e., the additive and multiplicative operations on \(V\) coincide and thus \(V\) is simply a group. It turns out that this group is very special.

Further, two particular automorphisms of \(V\) will play a crucial role. Namely, as \(V\) is invariant with respect to conjugation by \(x\) and \(\lambda\), it follows that there exist automorphisms \(A,A'\in\mathop{\mathrm{Aut}}(V)\) such that \[x+v-x=Av\quad\text{and}\quad\lambda(v)=A'v\] for each \(v\in V\) (i.e., \(A\) and \(A'\) are restrictions of \(\sigma_{-x}\) and \(\lambda\), respectively, to \(V\)). Moreover, let \[\mathcal{A}=\langle A,A' \rangle\] denote the subgroup of \(\mathop{\mathrm{Aut}}(V)\) generated by \(A\) and \(A'\).

Also the following observation will be useful.

Next, notice that the fourth condition in part (2) of Theorem 1 says that the action of \(V\) on \(X\) is transitive. Since \(\lambda_v=\mathop{\mathrm{id}}\) for all \(v\in V\), this means that the action of \(\mathcal{C}=\langle \sigma_v:v\in V \rangle\) on \(X\) is transitive. That is, the (additive) conjugation action of \(V\) on \(X\) is transitive. So, \[X=\mathcal{C}_{(B,+)}(x)=\mathcal{C}_{(V,+)}(x)=\{v+x-v:v\in V\}=\{v-Av+x:v\in V\}.\]

Further, since \(X\) is \(\lambda\)-invariant, we get \[\lambda(x)=u_0+x-u_0\] for some \(u_0\in V\) (note that the element \(u_0\) is in general not uniquely determined, but since \(u_0-Au_0=\lambda(x)-x=\lambda_x(x)-x=x*x\), the element \(u_0\) is unique as long as \(A\) has no non-zero fixed points).

Define \[v_n=u_0-A^nu_0\in V\] for \(n\geqslant 0\). Note that \(v_0=0\) and \(v_n=v_1+Av_1+\dotsb+A^{n-1}v_1\) for \(n\geqslant 1\). Moreover, \(\lambda(x)=v_1+x\). Now, if \(v\in V\) and \(0\leqslant i<k\) then using the above notation we can write \[\begin{align} \label{lam} \begin{aligned} \lambda(v+ix) & =\lambda(v)+i\lambda(x)=A'v+i(u_0+x-u_0) =A'v+u_0+ix-u_0\\ &=A'v+u_0-A^iu_0+ix=A'v+v_i+ix. \end{aligned} \end{align}\tag{7}\] Furthermore, \[\begin{align} v_1+AA'v+x & =v_1+x+A'v=\lambda(x)+\lambda(v)=\lambda(x+v)\\ & =\lambda(Av+x)=\lambda(Av)+\lambda(x)=A'Av+v_1+x. \end{align}\] Therefore, \(v_1+AA'v=A'Av+v_1\). So, replacing \(v\) by \(A^{-1}A'^{-1}v\) we may rewrite the last equality as \(AA'A^{-1}A'^{-1}v=-v_1+v+v_1\). Finally, since \(v\in V\) is arbitrary the latter equality is equivalent to \[[A,A']=\sigma_{v_1}\text{ on }V.\label{C2}\tag{8}\]

Our further investigation naturally splits into two cases (depending whether \(V\) is abelian or non-abelian), which nevertheless shares some similarities. We first treat the abelian case.

5.1 The abelian case↩︎

So, assume that the group \(V\) is abelian. In this case Lemma 13 implies that \(V\cong C_p^n\). Moreover, since \(kx\in V\), we get \(kx\in V\cap Z(B,+)=0\) by Lemma 15(2). Hence, \(\langle x \rangle_+\cong C_k\) and \((B,+)\cong V\rtimes\langle x \rangle_+\cong C_p^n\rtimes C_k\).

Moreover, we have \(A-1\in\mathop{\mathrm{Aut}}(V)\) (here and later we denote the identity map \(\mathop{\mathrm{id}}_V\) of \(V\) as \(1\), for simplicity). Indeed, if \(Av=v\) for some \(0\ne v\in V\) then \(x+v-x=Av=v\), and thus \(v\in V\cap Z(B,+)=0\) by Lemma 15(3), a contradiction.

Further, \(kx=0\) implies that \(A^k=1\quad\text{and}\quad A'^k=1\), and thus the (multiplicative) orders \(o(A)\) and \(o(A')\) of \(A\) and \(A'\), respectively, divide \(k\). Furthermore, we claim that \[\label{OrdCond} A^i\ne 1\text{ or }A'^i\ne 1\text{ or }1+A'+\dotsb+A'^{i-1}\ne 0\text{ for each }0<i<k.\tag{9}\] Indeed, suppose \(A^i=A'^i=1\) and \(1+A'+\dotsb+A'^{i-1}=0\) for some \(0<i<k\). Then \(ix\in Z(B,+)\) by Lemma 15(1). Because of 7 we have \(\lambda(ix)=u_0-A^iu_0+ix=ix\). In particular, the subgroup \(I=\langle ix \rangle_+\subseteq Z(B,+)\) is \(\lambda\)-invariant. Next, by an easy induction argument and because \(1+A'+\dotsb+A'^{i-1}=0\), we obtain \[\lambda^i(x) =\lambda^{i-1}(v_1+x)=A'^{i-1}v_1+\lambda^{i-1}(x)=A'^{i-1}v_1+\dotsb+A'v_1+v_1+x=x.\] Hence, \(\lambda^i(v+jx)=\lambda^i(v)+j\lambda^i(x)=A'^iv+jx=v+jx\) for all \(v\in V\) and \(0\leqslant j<k\). Therefore, \(\lambda^i=\mathop{\mathrm{id}}\) and, in consequence, \((ix)*b=\lambda_{ix}(b)-b=\lambda^i(b)-b=0\) for each \(b\in B\), i.e., \(I*B=0\). Summarising, \(I\) is a non-zero ideal of \(B\), which clearly does not contain \(V\), a contradiction.

Finally, since \(\sigma_{v_1}=\mathop{\mathrm{id}}\) on \(V\), as \(V\) is abelian, the condition 8 yields \(AA'=A'A\). In particular, the group \(\mathcal{A}=\langle A,A' \rangle\) is abelian. Furthermore, if \(\mathcal{R}\) denotes the subring of the endomorphism ring \(\mathop{\mathrm{End}}(V)\) generated by \(A\) and \(A'\) then Lemma 14 assures that \(V\) is a simple \(\mathcal{R}\)-module. A useful remark is that, since \(V*B=0\), an ideal of \(B\) contained in \(V\) is a synonym for \(\mathcal{R}\)-submodule of \(V\).

Therefore, we have proved the last part of the following result which is a refinement of Theorem 1(2) in case \(B^{(3)}=0\) and \(V\) is abelian.

Although Theorem 17 gives many examples, we provide a concrete construction that, under certain extra conditions, yields solutions of this type.

5.2 The non-abelian case↩︎

Now, let us assume that \(V\) is non-abelian. As we already know (see Lemma 13), in this case \(V=S_1+\dotsb+S_n\), a direct sum of isomorphic non-abelian simple groups, say isomorphic with \(S\).

Further, Remark 11 guarantees that \(B=V+C\), where \(C=\langle x \rangle_+\), a cyclic group of order \(m\), for some \(m\). Since \(B/V\cong\langle x+V \rangle_+\), we clearly get that \(k=|B/V|\) divides \(m\). Since \(V\ne B\), we get \(k>1\), as before. However, in contrast to the case in which \(V\) is abelian, it may happen that \(kx\ne 0\) and thus \(V\cap C\ne 0\).

Let \(D=V\rtimes C\) be the semi-direct product of the additive groups \(V\) and \(C\), where the action of \(C\) on \(V\) is determined by the automorphism \(A\) (introduced just before Lemma 14), i.e., \((v,ix)+(w,jx)=(v+A^iw,(i+j)x)\) for \((v,ix)\in D\) and \((w,jx)\in D\). We claim that \((B,+)\cong D/Z(D)\) Indeed, taking into account that \[Z(D)=\{(v,ix)\in D:Av=v\text{ and }A^iw=-v+w+v\text{ for each }w\in V\},\] it is easy to verify that the map \(\phi\colon D\to B\), defined as \(\phi(v,ix)=v+ix\), is a surjective morphism of groups with \(\mathop{\mathrm{Ker}}(\phi)=Z(D)\). Hence, \(\phi\) induces the desired isomorphism.

Next, observe that \[A'(kx)=\lambda(kx)=k\lambda(x)=k(u_0+x-u_0)=u_0+kx-u_0=u_0-A^ku_0+kx=v_k+x.\]

Finally, we claim that \(V=\langle v-Av:v\in V \rangle_+\), that is \(V\) is additively generated by elements of the form \(v-Av\) with \(v\in V\). Moreover, as we have \(v-Av=y-x\) for \(y=v+x-v\in X\), our claim may be equivalently expressed as that \(V\) is additively generated by elements of the form \(y-x\) with \(y\in X\) or, by elements of the form \(y-z=(y-x)-(z-x)\) with \(y,z\in X\). To prove the claim put \(I=\langle y-z:y,z\in X \rangle_+\). Since the set \(X\) is a conjugacy class that is both \(A\)-invariant and \(A'\)-invariant (and thus \(\mathcal{A}\)-invariant), it follows that \(I\) is an \(\mathcal{A}\)-invariant normal subgroup of \((V,+)\). Since \(|X|>1\), we get \(I\ne 0\) and thus \(I=V\) by Lemma 14, as claimed.

Therefore, we have proved the necessity part of the following refinement of Theorem 1(2) in case \(B^{(3)}=0\) and \(V\) is non-abelian.

We know that if \(B^{(2)}=0\) then \(V=[B,B]\), and \(V=B^{(2)}\) otherwise. Thus, in both cases, \(V\) is uniquely determined by the isomorphism class of the skew left brace \(B\). Hence, the number \(k=|B/V|\) is uniquely determined and, by the Jordan–Hölder theorem, also the number \(n\) of simple factors of \(V\) is uniquely determined.

Skew left braces \(B\) as in Theorem 17 and Theorem 20 are determined by a simple group \(S\), a positive integer \(n\) (this data determines the additive group \(V\), and \(V\) is abelian if and only if the simple group \(S\) is cyclic), an integer \(k>1\), two elements \(A\) (determining conjugation by \(x\) on \(V\)) and \(A'\) (determining restriction of \(\lambda\) to \(V\)) in \(\mathop{\mathrm{Aut}}(V)\), an element \(x\in B\) such that \(x+V\in B/V\) is an element of order \(k\), and an element \(u_0\in V\) (also needed to determine \(\lambda\)).

6 Examples↩︎

In the first four examples of this section we include some concrete classes of skew braces that satisfy all the requirements of the main theorems. In the fifth example we show that our results can be applied to simple skew left braces to generate further families of simple solutions. And we finish with some remarks on simple solutions of prime cardinality.

Recall that any finite group \((G,\circ)\) can be considered as a skew left brace \((G,\circ,\circ)\), the trivial skew left brace on the group \(G\), in particular \(G^{(2)}=0\). Note that in this skew left brace \(\lambda_a=\mathop{\mathrm{id}}\) and \(\rho_a=\sigma_a\) is the conjugation by \(a^{-1}\). Hence, the associated solution is \(r_G(a,b)=(b, b^{-1}\circ a \circ b)\) is a left derived solution, called the left conjugation solution. As mentioned earlier, in such a trivial skew left brace the ideals are precisely the normal subgroups of \((G,\circ)\). Of course in general one has several skew left brace structures on \(G\). However, as a consequence on Hopf–Galois structures, Byott proves in [50] that if \((G,\circ)\) is a non-abelian simple group then there are only two possible skew left brace structures on \(G\), namely the trivial skew left brace \((G,\circ, \circ)\) and the almost trivial skew left brace \((G,\circ^{\mathop{\mathrm{op}}},\circ)\), where \(\circ^{\mathop{\mathrm{op}}}\) is the opposite operation of \(\circ\). So, in this case we have \(G^{(2)}=G\). The latter corresponds with the non-degenerate solutions \((X,r)\) with all \(\rho_x=\mathop{\mathrm{id}}\) and thus \(\sigma_x=\lambda^{-1}_x\) for all \(x\in X\). In this case the associated solution is \(r_G(a,b)=(a\circ b \circ a^{-1},a)\), called the right conjugation solution.

We end with a general remark on solutions of prime order.

This can easily be proven. First, we say that a morphism \(f\colon(X,r)\to(Y,s)\) of solutions is a covering map if it is surjective and all the fibres \(f^{-1}(y)\) for \(y\in Y\) have the same cardinality. A non-degenerate solution \((X,r)\) is said to be transitive if \(\mathcal{S}(X,r)\) acts transitively on \(X\). By[51] (for non-degenerate involutive solutions) and by [45] (for the general case), if \(f\colon(X,r)\to(Y,s)\) is an epimorphism of solutions then \(f\) is a covering map. In particular, if \(|X|\) is prime then \((X,r)\) is a simple solution.

In [47] Etingof, Guralnick, and Soloviev have described indecomposable finite non-degenerate solutions \((X,r)\) of the Yang–Baxter equation that are of prime order. It turns out that then \(X\) can be equipped with an abelian group structure \((X,+)\) (and thus we may assume that \(X=\mathbb{Z}_p\)) such that the solution \((X,r)\) is affine, i.e., it is of the form \[r(x,y)=(\alpha(x)+\beta(y)+a,\gamma(x)+\delta(y)+b)\] for some \(\alpha,\beta,\gamma,\delta\in\mathop{\mathrm{End}}(X,+)\) and \(a,b\in X\). They are precisely the solutions of the following type built on triples \(a,b,c\in X\) with \(a,b\ne 0\) such that \(ab\ne 1\), and pairs \(c_1,c_2\in X\) such that \((c_1,c_2)\ne(0,0)\):

1. \((x,y)\mapsto (ay+(1-ab)x+c,bx-a^{-1}c)\),

2. \((x,y)\mapsto (ay+c,bx+(1-ab)y-bc)\),

3. \((x,y)\mapsto (y+c_1,x+c_2)\).

The solutions of the third type are the solutions described in Proposition 9 and the solutions of the first and second type are solutions determined by a simple skew left brace \(B\).

It is interesting to note that all these solutions satisfy \(\lambda_x=\lambda_y\) for all \(x,y\in X\) or \(\rho_x=\rho_y\) for all \(x,y\in X\), and thus are of the type investigated in [49], where it is shown that the Yang–Baxter algebras corresponding to such solutions always are left, respectively right, Noetherian.

Acknowledgement↩︎

The first named author is partially supported by the Fonds voor Wetenschappelijk Onderzoek (Flanders) – Krediet voor wetenschappelijk verblijf in Vlaanderen grant V512223N. The third author is supported by National Science Centre grant 2020/39/D/ST1/01852 (Poland). The fourth author is supported by Fonds Wetenschappelijk Onderzoek Vlaanderen (grant 1229724N).

References↩︎