Relations between crossed modules of different algebras

Abstract

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 in 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 [d_1,d_2]= d_1 -| d_2 - d_2 |- d_1, for all d_1, d_2 in D, 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.