Representability of actions in the category of (Pre)crossed modules in Leibniz algebras

2016 ◽  
Vol 45 (5) ◽  
pp. 1825-1841 ◽  
Author(s):  
M. Atık ◽  
A. Aytekın ◽  
E.Ö. Uslu
Keyword(s):  
Leibniz Algebras ◽  
Crossed Modules

scholarly journals Braiding for categorical algebras and crossed modules of algebras II: Leibniz algebras

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1443-1469
Author(s):  
Alejandro Fernández-Fariña ◽  
Manuel Ladra
Keyword(s):  
Lie Algebras ◽  
Leibniz Algebras ◽  
Crossed Modules

In this paper, we study the category of braided categorical Leibniz algebras and braided crossed modules of Leibniz algebras, and we relate these structures with the categories of braided categorical Lie algebras and braided crossed modules of Lie algebras using the Loday-Pirashvili category.

scholarly journals From simplicial homotopy to crossed module homotopy in modified categories of interest

2020 ◽  
Vol 27 (4) ◽  
pp. 541-556
Author(s):  
Kadir Emir ◽  
Selim Çetin
Keyword(s):  
Lie Algebras ◽  
Equivalence Relation ◽  
Crossed Module ◽  
Algebraic Structures ◽  
Homotopy Equivalence ◽  
Associative Algebras ◽  
Leibniz Algebras ◽  
Crossed Modules ◽  
Simplicial Objects ◽  
Simplicial Homotopy

AbstractWe address the (pointed) homotopy of crossed module morphisms in modified categories of interest that unify the notions of groups and various algebraic structures. We prove that the homotopy relation gives rise to an equivalence relation as well as to a groupoid structure with no restriction on either domain or co-domain of the corresponding crossed module morphisms. Furthermore, we also consider particular cases such as crossed modules in the categories of associative algebras, Leibniz algebras, Lie algebras and dialgebras of the unified homotopy definition. Finally, as one of the major objectives of this paper, we prove that the functor from simplicial objects to crossed modules in modified categories of interest preserves the homotopy as well as the homotopy equivalence.

scholarly journals More on crossed modules in Lie, Leibniz, associative and diassociative algebras

2017 ◽  
Vol 16 (06) ◽  
pp. 1750107 ◽  
Author(s):  
J. M. Casas ◽  
R. F. Casado ◽  
E. Khmaladze ◽  
M. Ladra
Keyword(s):  
Associative Algebras ◽  
Adjoint Functors ◽  
Leibniz Algebras ◽  
Crossed Modules

Adjoint functors between the categories of crossed modules in dialgebras and Leibniz algebras are constructed. The well known relations between the categories of Lie, Leibniz, associative algebras and dialgebras are extended to the respective categories of crossed modules.

On solvable Leibniz algebras whose nilradical has characteristic sequence (m1,m2,m3)

2018 ◽  
Vol 2018 (3) ◽  
pp. 4-17
Author(s):  
K.K. Abdurasulov ◽  
Drew Horton ◽  
U.X. Mamadaliyev
Keyword(s):  
Characteristic Sequence ◽  
Leibniz Algebras

scholarly journals Abelian extensions and crossed modules of Hom-Lie algebras

2020 ◽  
Vol 224 (3) ◽  
pp. 987-1008
Author(s):  
José Manuel Casas ◽  
Xabier García-Martínez
Keyword(s):  
Lie Algebras ◽  
Abelian Extensions ◽  
Crossed Modules

Crossed Semimodules and Schreier Internal Categories in the Category of Monoids

1998 ◽  
Vol 5 (6) ◽  
pp. 575-581 ◽  
Author(s):  
A. Patchkoria
Keyword(s):  
Internal Category ◽  
Equivalence Of Categories ◽  
Crossed Modules

Abstract We introduce the notion of a Schreier internal category in the category of monoids and prove that the category of Schreier internal categories in the category of monoids is equivalent to the category of crossed semimodules. This extends a well-known equivalence of categories between the category of internal categories in the category of groups and the category of crossed modules.

scholarly journals Naturally Graded 2-Filiform Leibniz Algebras

2010 ◽  
Vol 38 (10) ◽  
pp. 3671-3685 ◽  
Author(s):  
L. M. Camacho ◽  
J. R. Gómez ◽  
A. J. González ◽  
B. A. Omirov
Keyword(s):  
Leibniz Algebras

scholarly journals Computation and homotopical applications of induced crossed modules

2003 ◽  
Vol 35 (1) ◽  
pp. 59-72 ◽  
Author(s):  
Ronald Brown ◽  
Christopher D. Wensley
Keyword(s):  
Crossed Modules
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