Skew braces and their connection to the Yang-Baxter equation, Hopf-Galois structures and post-Lie rings

The Yang–Baxter equation initially appeared in both quantum and statistical mechanics. Drinfeld proposed studying a special class of its solutions, the so-called set-theoretical solutions. Although more restrictive, these solutions remain too general to be classified completely. Nevertheless, studying them leads to a better
understanding of the Yang–Baxter equation. Braces, algebraic structures introduced by Rump, turn out to provide the right algebraic framework to study involutive non-degenerate solutions. Later, skew braces were introduced by Guarnieri and Vendramin as a generalization to study also the non-involutive ones. These algebraic structures form the main focus of this thesis.

We start by studying two classes of skew braces. The first is the class of two-sided skew braces. We show that they can be described as extensions of weakly trivial skew braces by two-sided braces, which leads to novel structural results. The second class comprises bi-skew braces, which are more generally related to brace blocks. We provide a technical characterization of brace blocks as well as more feasible constructions.

In the next three chapters, we treat the connection between skew braces and non-degenerate set-theoretical solutions of the Yang–Baxter equation. After exploring how certain properties of solutions relate to those of their associated skew braces, we classify finite indecomposable involutive non-degenerate solutions with
an abelian permutation group and indecomposable involutive non-degenerate solutions of size p 2 with p a prime.

Subsequently, we discuss the existing connection between skew braces and Hopf–Galois structures on finite Galois field extensions and propose a refined version. This enables an explicit classification of all such extensions where the Hopf–Galois correspondence is surjective, a behavior closely resembling that of the classical Galois correspondence.

In the final part of the thesis, we develop a Lazard correspondence between L-nilpotent post-Lie rings and L-nilpotent skew braces. By means of illustration, we then use this correspondence to obtain in a more explicit form Zenouz’s classification of skew braces of order p 3, p > 3 a prime, through a classification of L-nilpotent Lie rings of the same order.

Collectively, these results deepen the structural understanding of skew braces and emphasize their pivotal role, both in algebra and in relation to the Yang–Baxter equation.