Moreno Invitti - Skew Braces of Finite Morley Rank
112
ICJ
Skew braces are a class of algebraic structures introduced by Guarnieri and Vendramin to study set-theoretic solutions of the Yang–Baxter equation. A skew brace consists of a set equipped with two group operations satisfying a compatibility condition known as skew-left distributivity. This framework bridges group theory, ring theory, and mathematical physics, and has attracted growing interest from an algebraic perspective in recent years. Notions such as solvability and nilpotency - particularly (strong) left nilpotency - have been developed and explored within this context.
In this talk, we investigate skew braces under the assumption of finite Morley rank, a model-theoretic concept that generalizes the idea of
dimension from algebraic geometry. In particular, we present a complete classification of skew braces of Morley rank at most 3. Additionally, we prove that if both the additive and multiplicative groups of a skew brace are nilpotent, then the skew brace is strong left
nilpotent. Finally, under an assumption concerning the length of chains of left ideals, we show that if both the additive and multiplicative
groups are solvable, then the skew brace is weakly solvable.