A \emph{skew brace} is a triple $(B,+,\circ),$ where $(B,+)$ and $(B,\circ)$ are groups such that for all $a,b,c \in B$ a skew left distributivity holds $$a\circ(b+c)=a\circ b-a+a\circ c,$$ where $-a$ denotes the inverse of $a$ in $(B,+)$. Given a skew brace $(B,+,\circ)$, there is an action by automorphisms of $(B,\circ)$ on $(B,+)$ defined by
$$\lambda_a(x)=-a+a\circ x$$
and an action by automorphisms of $(B,+)\rtimes_{\lambda} (B,\circ)$ on $(B,+)$ defined by
$$\theta_{(a,b)}(c)=a+\lambda_b(c)-a.$$
In this talk, we will see how the group structures $(B,+)$ and $(B,\circ)$ are related when each element of $B$ has a finite number of $\theta$-images. We will observe that this topic is closely related to the concept of $FC$-group, introduced in the context of Group Theory by R. Baer.
In the second part, we relate our results to non-degenerate set-theoretic solutions. A set-theoretic solution to the Yang-Baxter equation is a tuple $(X,r)$ such that $X$ is a non-empty set and $r\colon X \times X \rightarrow X \times X$ is a bijective map such that $$ (r \times \id_X)(\id_X \times r)( r \times \id_X) = (\id_X \times r)(r \times \id_X)(\id_X\times r).$$ Denote $r(x,y) = (\lambda_x(y),\rho_y(x))$. If $\lambda_x,\rho_x$ are bijective for all $x \in X$, then one says that $(X,r)$ is non-degenerate.
In this talk, we establish an analogue of Sylow’s Third Theorem for finite skew braces. We prove that, as in the classical group-theoretic setting, the number of Sylow p-sub-skew braces is always congruent to 1 (mod p). On the other hand, a key feature of the classical theorem does not extend to skew braces: the number of Sylow p-sub-skew braces need not divide the order of the skew brace. This highlights a significant difference between the Sylow theory of groups and its counterpart for skew braces.
This talk is based on recent work in collaboration with E. Jespers, T. Letourmy, M. Trombetti and A. Van Antwerpen on free skew braces in certain classes and their relation to set-theoretic solutions to the Yang Baxter equation. We present a construction of free right nilpotent skew braces of class n. This yields a concrete realization of the free object and allows us to derive new structural properties. As an illustration, we discuss the one-generated case and its relation to free solutions in this setting.
Recent developments in the study of set-theoretic Yang--Baxter equation (YBE) have highlighted the key role played by semigroup theory. In particular, right groups have turned out to be effective for analyzing different classes of solutions.
The aim of this talk is to explore some recent connections discovered between structures related to right groups and the YBE. Specifically, we show how solutions obtained in this way belong to a broader class of solutions that properly contains skew braces solutions.
In this talk I will discuss various aspects of the category MonFun(B,Mat) whose objects are braid representations, i.e. strict monoidal functors from the braid category B to the category of matrices Mat. The objects of this category are equivalent to solutions to the constant Yang–Baxter equation. The morphisms in MonFun(B,Mat) are the monoidal natural transformations between braid representations.
This categorical approach allows us to frame attempts at classification in terms of restricting the source B, the target Mat, and in terms of subcategories of MonFun(B,Mat). Our approach was particularly motivated by the recent successful classification of charge-conserving braid representations. Our aims are to understand how the solution was facilitated by the restriction with a view to generalisation, and to understand how universal various restrictions are, in terms of some notion of equivalence. We will state several results on the structure of subcategories of MonFun(B,Mat) as well as highlight some open questions.
Drinfeld formulated the set-theoretic Yang Baxter equation (YBE) $$(r\times 1_Q)(1_Q\times r)(r\times 1_Q)=(1_Q\times r)(r\times 1_Q)(1_Q\times r)$$ for a map $r:Q^2\to Q^2$, in an attempt to simplify the classification problem for solutions of the quantum YBE from physics. In the intervening 35 years, investigations of the set-theoretic YBE have called upon a wide array of algebraic structures: groups, quasigroups, racks, quandles, cycle sets, and (skew)-braces, just to name a few.
The structure groups of the solutions to the set-theoretic YBE carry left-brace structures (specific ring-like structures introduced by Rump in 2005). For finite, nondegenerate, and involutive solutions, the corresponding structure groups are torsion-free crystallographic groups, and the additive structure of the associated braces is a free abelian group. Recently, using the properties of these objects, Rump extended left-brace theory to the class of crystallographic groups. He obtained in this way \emph{cofinite integral braces}.
In this talk, we will show that on any crystallographic group, only finitely many cofinite integral braces, up to isomorphism, can exist, and we will present a method for determining them, together with some computational results. This is joint work with Rafa{\l} Lutowski and Andrzej Szczepa\'{n}ski from University of Gda\'{n}sk.