Braided Crossed Modules and Loday-Pirashvili category

Abstract

This thesis is devoted to the study of braidings in different mathematical contexts, as well as in a deeper analysis of the Loday-Pirashvili category. We will study the notion of braidings for crossed modules and internal categories in the cases of groups, associative algebras, Lie algebras and Leibniz algebras, showing the equivalence between the respective categories. We will also study universal central extensions in the category of braided crossed modules of Lie algebras. Finally, we will show how to generalize the Loday-Pirashvili category. With that construction, we will exhibit a generalization of the relationship between Lie and Leibniz objects.