Relations between crossed modules of different algebras

In the present work we extend to crossed modules the classical adjunction between the Liezation functor Liea : As Lie, which makes every associative algebra A into a Lie algebra via the bracket a,b ab ba , for all a,b A, and U : Lie As, which assigns to every Lie algebra p its universal enveloping algebra U( p ). Likewise, we construct a 2 dimensional generalization of the adjunction between the functor Lb : Di Lb, which assigns to every dialgebra D the Leibniz bracket given by 1 2 1 d ,d d ┤ 2 2 d d ├ 1 d , for all d d D 1 2 , , and Ud : Lb Di, the universal enveloping dialgebra functor. Additionally, we assemble all the resulting squares of categories and functors in four parallelepipeds, for which, in every face, the inner and outer squares are commutative or commute up to isomorphism. Since our second generalization involves crossed modules of dialgebras, we give an adequate definition for them, based on the more general notion of crossed modules in categories of interest. Furthermore, we define the concept of strict 2 dialgebra, by analogy to the notion of strict associative 2 algebra. We prove that the categories of crossed modules of dialgebras and strict 2 dialgebras are equivalent. Additionally, we construct the dialgebra of tetramultipliers, which happens to be the actor in the category of dialgebras under certain conditions. Besides, given a Leibniz crossed module, we construct a general actor crossed modules, which is the actor in some particular cases

Categorical-algebraic methods in non-commutative and non-associative algebra

The objective of this dissertation is twofold: firstly to use categorical and algebraic methods to study homological properties of some of the aforementioned semi-abelian, non-associative structures and secondly to use categorical and algebraic methods to study categorical properties and provide categorical characterisations of some well-known algebraic structures. On one hand, the theory of universal central extensions together with the non-abelian tensor product will be studied and used to explicitly calculate some homology groups and some problems about universal enveloping algebras and actions will be solved. On the other hand, we will focus on giving categorical characterisations of some algebraic structures, such as a characterisation of groups amongst monoids, of cocommutative Hopf algebras amongst cocommutative bialgebras and of Lie algebras amongst alternating algebras