# ALGB Seminar: Ilaria Colazzo (Università del Salento)

Title: Regular subgroups and left semi-braces

Abstract: The Yang-Baxter equation is one of the fundamental equations in mathematical-physics. The problem of finding and classifying the set theoretical solutions of the Yang-Baxter equation has originally been posed by Drinfeld in [5]. Rump introduced braces, as a generalization of radical rings in [8], in order to construct set-theoretical involutive solutions of the Yang-Baxter equation. Braces over a field have an unexpected connection with regular subgroups of an affine group, as discovered in [3]. Finding all regular subgroups of an affine group is an open problem formalized by Liebeck, Praeger and Saxl in 2010 (see [7]). Last year Guarnieri and Vendramin [6] obtained a generalization of braces, namely the skew braces, and Catino, Colazzo and Stefanelli [1] obtained the semi-braces as a further generalization. In this talk we first explore the link between braces over a field a regular subgroups of the affine group and we prove that with this linkage we can construct a family of regular subgroups of the affine group such that they have trivial intersection with the translation group [2]. Then we extend this result to the connection between skew braces and regular subgroups of the holomorph of a group, as obtained in [6]. Finally, we introduce a suitable definition of the holomorph of a right group and extend the link between semi-braces and regular subgroups of this holomorph [4].