Conexiones de raíces en estructuras split

Abstract

In the Thesis it is exhibited a study of multiple split structures in several classes of nonassociative algebras and superalgebras, based on the pioneering work of Dynkin and Cartan. The knowledge of this kind of decomposition for each class of algebras, allow us to conclude some important facts about its structure and representations. Generalizing what is known for Lie algebras and Lie superalgebras, we begin our research fixing an adequate subalgebra and after, defining the concepts of root spaces and in this set an equivalence relation, we reach for a decomposition of the algebra as a direct sum of some special ideals that, under certain conditions, are simple algebras (superalgebras). In this work, the classes of Lie color algebras, Leibniz superalgebras and Malcev algebras are studied. A last appendix is presented where the techniques are adapted to graded structures defined in Lie superalgebras and Leibniz algebras. In general, this Thesis is a study where a lot of original results are proved, trying to contribute for the development of the theory of non-associative algebras.