An irreducible component of the variety of Leibniz algebras having trivial intersection with the variety of Lie algebras

2013 ◽  
Vol 62 (11) ◽  
pp. 1450-1459 ◽  
Author(s):  
J.M. Ancochea Bermúdez ◽  
R. Campoamor-Stursberg
Keyword(s):  
Irreducible Component ◽  
Lie Algebras ◽  
Leibniz Algebras ◽  
Trivial Intersection

Some new results on Gröbner–Shirshov bases for Lie algebras and around

2018 ◽  
Vol 28 (08) ◽  
pp. 1403-1423
Author(s):  
L. A. Bokut ◽  
Yuqun Chen ◽  
Abdukadir Obul
Keyword(s):  
Lie Algebras ◽  
Associative Algebras ◽  
Leibniz Algebras ◽  
Novikov Algebras

We review Gröbner–Shirshov bases for Lie algebras and survey some new results on Gröbner–Shirshov bases for [Formula: see text]-Lie algebras, Gelfand–Dorfman–Novikov algebras, Leibniz algebras, etc. Some applications are given, in particular, some characterizations of extensions of groups, associative algebras and Lie algebras are given.

scholarly journals Leibniz Algebras Constructed by Representations of General Diamond Lie Algebras

2017 ◽  
Vol 42 (3) ◽  
pp. 1281-1293 ◽  
Author(s):  
L. M. Camacho ◽  
I. A. Karimjanov ◽  
M. Ladra ◽  
B. A. Omirov
Keyword(s):  
Lie Algebras ◽  
Leibniz Algebras

scholarly journals Leibniz algebras associated with representations of filiform Lie algebras

2015 ◽  
Vol 98 ◽  
pp. 181-195 ◽  
Author(s):  
Sh.A. Ayupov ◽  
L.M. Camacho ◽  
A.Kh. Khudoyberdiyev ◽  
B.A. Omirov
Keyword(s):  
Lie Algebras ◽  
Leibniz Algebras ◽  
Filiform Lie Algebras

scholarly journals Classification of 6D Leibniz algebras

2020 ◽  
Vol 2020 (7) ◽  
Author(s):  
Ladislav Hlavatý
Keyword(s):  
Lie Algebras ◽  
Algebraic Structure ◽  
Leibniz Algebras ◽  
Drinfel’D Double ◽  
Generalized Frame

Abstract Leibniz algebras ${\mathcal E}_n$ were introduced as an algebraic structure underlying U-duality. Algebras ${\mathcal E}_3$ derived from Bianchi 3D Lie algebras are classified here. Two types of algebras are obtained: 6D Lie algebras that can be considered an extension of the semi-Abelian 4D Drinfel’d double and unique extensions of non-Abelian Bianchi algebras. For all of the algebras explicit forms of generalized frame fields are given.

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.

Solvable Leibniz algebras with quasi-filiform Lie algebras of maximum length nilradicals

2020 ◽  
Vol 48 (8) ◽  
pp. 3525-3542
Author(s):  
Kh. A. Muratova ◽  
M. Ladra ◽  
B. A. Omirov ◽  
A. M. Sattarov
Keyword(s):  
Lie Algebras ◽  
Maximum Length ◽  
Leibniz Algebras ◽  
Filiform Lie Algebras

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 On split regular Hom-Leibniz algebras

2018 ◽  
Vol 17 (10) ◽  
pp. 1850185 ◽  
Author(s):  
Yan Cao ◽  
Liangyun Chen ◽  
Bing Sun
Keyword(s):  
Lie Algebras ◽  
Natural Generalization ◽  
Leibniz Algebra ◽  
Maximal Length ◽  
Abelian Subalgebra ◽  
Leibniz Algebras

We introduce the class of split regular Hom-Leibniz algebras as the natural generalization of split Leibniz algebras and split regular Hom-Lie algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular Hom-Leibniz algebra [Formula: see text] is of the form [Formula: see text] with [Formula: see text] a subspace of the abelian subalgebra [Formula: see text] and any [Formula: see text], a well described ideal of [Formula: see text], satisfying [Formula: see text] if [Formula: see text]. Under certain conditions, in the case of [Formula: see text] being of maximal length, the simplicity of the algebra is characterized.

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