PhD thesis

  • I. Colazzo, Left semi-braces and the Yang-Baxter equation, PhD thesis, University of Salento (2017).


Research interests

My research activity focuses on the study of the set-theoretical solutions of the Yang-Baxter equation introduced in [Drinfeld, 1992]. In 2005, Rump presented braces as a generalization of radical rings that turn out to be useful for studying involutive non-degenerate solutions using ring-theoretical and group-theoretical. From this perspective, in 2017 Guarnieri and Vendramin introduced skew braces as a generalization of braces that are in correspondence with non-degenerate bijective solutions which are not necessarily involutive of solutions. In 2017, in [F. Catino, I. Colazzo, P. Stefanelli: Semi-braces and the Yang-Baxter equation, J. Algebra (2017)] we introduced the notion of a semi-brace, i.e., an algebraic structure that covers skew braces and leads to left non-degenerate solutions. In particular, we describe the structural aspects of a semi-brace and provide a precise characterization of this structure. Moreover, we introduce suitable concepts of ideal and quotient structure of a semi-brace and give a generalization of the socle, a particular ideal, that includes the definition for (skew) braces. Furthermore, we provide a construction of semi-braces, the asymmetric product, a generalization of the asymmetric product of braces that involves classical tools such as Schreier’s extension of groups (not necessarily abelian) and group actions.
In addition to the approach based on algebraic structures, another method to get new families of solutions is using other ones. In [F. Catino, I. Colazzo, P. Stefanelli: The matched product of set-theoretical solutions of the Yang-Baxter equation, submitted] we introduce a novel construction technique which allows obtaining new solutions on the cartesian product of sets, starting from completely arbitrary solutions. Strongly linked to the study of braces is the connection with regular subgroups. One of the main problems is determining regular subgroups of the affine group of a vector space; a still open problem formally posed in [Liebeck, Praeger, Saxl 2010]. Using the relationship between regular subgroups and radical F-braces found in [Catino, Rizzo 2009], it is possible to obtain in a systematic way such type of regular subgroups. In [F. Catino, I. Colazzo, P. Stefanelli: On regular subgroups of the affine group, Bull. Aust. Math. Soc., 91 (2015), 76-85], we translate cocycles and Hochschild products from the context of associative algebras into that of F-braces. Using Hochshild product we describe all finite-dimensional radical F-braces with non-trivial annihilator.
Moreover, in [F. Catino, I. Colazzo, P. Stefanelli: Regular subgroups of the affine group and asymmetric product of radical braces, J. Algebra, 455, (2016), 164–182.] we introduce a new construction of radical braces that extends the semidirect product of two radical braces. In particular, this construction technique allows us to generalize the regular subgroups constructed in [Hegedűs 2000] that have a trivial intersection with the translation group.