The reciprocity between solutions of the Yang-Baxter equation, skew braces and Lie algebras, with attention to computational applications

In this project I propose to study set-theoretic solutions of the Yang-Baxter equation and associated algebraic structures. In particular, I will examine several minimal building blocks of set-theoretic solutions using their respective structure groups and monoids. Secondly, as finite bijective non-degenerate solutions have an associated finite skew left brace, the permutation group, I study the important class of braces of prime power order. Indeed, as every skew left brace can be factorized into such skew braces and factorizations correspond to decompositions of solutions, these braces contain a lot of information on the complete brace/ solution. To aid in this study, I will employ the diverse notions of nilpotency of skew brace. Furthermore, as the Köthe conjecture can be naturally formulated using two-sided braces, we use these novel tools to analyze this conjecture. A seperate approach is the recently uncovered link with pre-Lie algebras, using this method, promises an influx of new techniques to attack this particular class of braces of prime power order. Where possible, I will link the newly developed theory for skew braces to results for their corresponding solutions. Last, using the development of techniques for skew left braces and set-theoretic solutions, I propose novel approaches using graph theory and probability theory to uncover more combinatorial data, which will aid in the efforts to build a classification of solutions and skew left braces of small order