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

SEMI-COMPLETE CROSSED MODULES OF LIE ALGEBRAS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250096
Author(s):  
A. AYTEKIN ◽  
J. M. CASAS ◽  
E. Ö. USLU
Keyword(s):  
Lie Algebras ◽  
Necessary Conditions ◽  
Crossed Module ◽  
Crossed Modules ◽  
Sufficient And Necessary Conditions

We investigate some sufficient and necessary conditions for (semi)-completeness of crossed modules in Lie algebras and we establish its relationships with the holomorphy of a crossed module. When we consider Lie algebras as crossed modules, then we recover the corresponding classical results for complete Lie algebras.

Derivation Hom-Lie 2-algebras and non-abelian extensions of regular Hom-Lie algebras

2018 ◽  
Vol 17 (05) ◽  
pp. 1850081 ◽  
Author(s):  
Lina Song ◽  
Rong Tang
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Homotopy Classes ◽  
Abelian Extensions ◽  
Isomorphism Classes ◽  
One To One

In this paper, we introduce the notion of a derivation of a regular Hom-Lie algebra and construct the corresponding strict Hom-Lie 2-algebra, which is called the derivation Hom-Lie 2-algebra. As applications, we study non-abelian extensions of regular Hom-Lie algebras. We show that isomorphism classes of diagonal non-abelian extensions of a regular Hom-Lie algebra [Formula: see text] by a regular Hom-Lie algebra [Formula: see text] are in one-to-one correspondence with homotopy classes of morphisms from [Formula: see text] to the derivation Hom-Lie 2-algebra [Formula: see text].

Representations and deformations of 3-Hom-ρ-Lie algebras

2021 ◽  
Author(s):  
Esmaeil Peyghan ◽  
Aydin Gezer ◽  
Zahra Bagheri ◽  
Inci Gultekin
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Abelian Extension ◽  
Abelian Extensions ◽  
Definition Of

The aim of this paper is to introduce 3-Hom-[Formula: see text]-Lie algebra structures generalizing the algebras of 3-Hom-Lie algebra. Also, we investigate the representations and deformations theory of this type of Hom-Lie algebras. Moreover, we introduce the definition of extensions and abelian extensions of 3-Hom-[Formula: see text]-Lie algebras and show that associated to any abelian extension, there is a representation and a 2-cocycle.

Perfect crossed modules in lie algebras

1995 ◽  
Vol 23 (5) ◽  
pp. 1625-1644 ◽  
Author(s):  
J. M. Casas ◽  
M. Ladra
Keyword(s):  
Lie Algebras ◽  
Crossed Modules

Colimits in the Crossed Modules Category in Lie Algebras

2000 ◽  
Vol 7 (3) ◽  
pp. 461-474 ◽  
Author(s):  
J. M. Casas ◽  
M. Ladra
Keyword(s):  
Lie Algebras ◽  
Left Adjoint ◽  
Crossed Modules

Abstract In this paper a left adjoint is constructed for the restriction (or pullback) functor associated to a morphism of Lie algebras. Colimits in the category of crossed modules in Lie algebras and a new result on crossed modules induced by a morphism of Lie algebras ι : P → Q in the case where ι is the inclusion of an ideal, are obtained.

Cohomology, derivations and abelian extensions of 3-Lie algebras

2019 ◽  
Vol 18 (07) ◽  
pp. 1950130 ◽  
Author(s):  
Senrong Xu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Representation Formula ◽  
Abelian Extension ◽  
Abelian Extensions ◽  
First Order

Given a representation [Formula: see text] of a 3-Lie algebra [Formula: see text], we construct first-order cohomology classes by using derivations of [Formula: see text], [Formula: see text] and obtain a Lie algebra [Formula: see text] with a representation [Formula: see text] on [Formula: see text]. In the case that [Formula: see text] is given by an abelian extension [Formula: see text] of 3-Lie algebras with [Formula: see text], we obtain obstruction classes for extensibility of derivations of [Formula: see text] and [Formula: see text] to those of [Formula: see text]. An application of the representation [Formula: see text] to derivations is also discussed.

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.

A Five-term Exact Sequence for Opext of Crossed Modules in Lie Algebras

2000 ◽  
Vol 7 (2) ◽  
pp. 121-138
Author(s):  
J. M. Casas ◽  
A. M. Vieites Rodríguez
Keyword(s):  
Exact Sequence ◽  
Lie Algebras ◽  
Crossed Modules
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