9 research outputs found
On crossed modules in modified categories of interest
We introduce some algebraic structures such as singularity, commutators and central extension in modified categories of interest. Additionally, we introduce the cat^{1} -objects and internal categories with
their connection to crossed modules in these categories, which gives rise to unification of many notions about (pre)crossed modules in various algebras of categories
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