Lie triple systems and Leibniz algebras

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Revaz Kurdiani
Keyword(s):  
Lie Algebra ◽  
Central Extension ◽  
Triple System ◽  
Leibniz Algebra ◽  
Universal Central Extension ◽  
Central Extensions ◽  
Lie Triple System ◽  
Triple Systems ◽  
Leibniz Algebras ◽  
Universal Central Extensions

AbstractThe present paper deals with the Lie triple systems via Leibniz algebras. A perfect Lie algebra as a perfect Leibniz algebra and as a perfect Lie triple system is considered and the appropriate universal central extensions are studied. Using properties of Leibniz algebras, it is shown that the Lie triple system universal central extension is either the universal central extension of the Leibniz algebra or the universal central extension of the Lie algebra.

scholarly journals On universal central extensions of Hom-Leibniz algebras

2014 ◽  
Vol 13 (08) ◽  
pp. 1450053 ◽  
Author(s):  
J. M. Casas ◽  
M. A. Insua ◽  
N. Pacheco Rego
Keyword(s):  
Lie Algebra ◽  
Central Extension ◽  
Chain Complex ◽  
Leibniz Algebra ◽  
Central Extensions ◽  
Leibniz Algebras ◽  
Leibniz Homology ◽  
Universal Central Extensions

In the category of Hom-Leibniz algebras we introduce the notion of Hom-co-representation as adequate coefficients to construct the chain complex from which we compute the Leibniz homology of Hom-Leibniz algebras. We study universal central extensions of Hom-Leibniz algebras and generalize some classical results, nevertheless it is necessary to introduce new notions of α-central extension, universal α-central extension and α-perfect Hom-Leibniz algebra due to the fact that the composition of two central extensions of Hom-Leibniz algebras is not central. We also provide the recognition criteria for these kind of universal central extensions. We prove that an α-perfect Hom-Lie algebra admits a universal α-central extension in the categories of Hom-Lie and Hom-Leibniz algebras and we obtain the relationships between both of them. In case α = Id we recover the corresponding results on universal central extensions of Leibniz algebras.

scholarly journals Universal central extensions of Lie–Rinehart algebras

2018 ◽  
Vol 17 (07) ◽  
pp. 1850134 ◽  
Author(s):  
J. L. Castiglioni ◽  
X. García-Martínez ◽  
M. Ladra
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Central Extension ◽  
Universal Central Extension ◽  
Central Extensions ◽  
Definition Of ◽  
Universal Central Extensions

In this paper, we study the universal central extension of a Lie–Rinehart algebra and we give a description of it. Then we study the lifting of automorphisms and derivations to central extensions. We also give a definition of a non-abelian tensor product in Lie–Rinehart algebras based on the construction of Ellis of non-abelian tensor product of Lie algebras. We relate this non-abelian tensor product to the universal central extension.

Some Properties of ID∗-n-Lie-derivations of Leibniz algebras

2021 ◽  
pp. 2250054
Author(s):  
G. R. Biyogmam ◽  
C. Tcheka ◽  
D. A. Kamgam
Keyword(s):  
Lie Algebra ◽  
Leibniz Algebra ◽  
Central Series ◽  
Algebra Structure ◽  
Leibniz Algebras ◽  
Lie Derivations

The concepts of [Formula: see text]-derivations and [Formula: see text]-central derivations have been recently presented in [G. R. Biyogmam and J. M. Casas, [Formula: see text]-central derivations, [Formula: see text]-centroids and [Formula: see text]-stem Leibniz algebras, Publ. Math. Debrecen 97(1–2) (2020) 217–239]. This paper studies the notions of [Formula: see text]-[Formula: see text]-derivation and [Formula: see text]-[Formula: see text]-central derivation on Leibniz algebras as generalizations of these concepts. It is shown that under some conditions, [Formula: see text]-[Formula: see text]-central derivations of a non-Lie-Leibniz algebra [Formula: see text] coincide with [Formula: see text]-[Formula: see text]-[Formula: see text]-derivations, that is, [Formula: see text]-[Formula: see text]-derivations in which the image is contained in the [Formula: see text]th term of the lower [Formula: see text]-central series of [Formula: see text] and vanishes on the upper [Formula: see text]-central series of [Formula: see text] We prove some properties of these [Formula: see text]-[Formula: see text]-[Formula: see text]-derivations. In particular, it is shown that the Lie algebra structure of the set of [Formula: see text]-[Formula: see text]-[Formula: see text]-derivations is preserved under [Formula: see text]-[Formula: see text]-isoclinism.

The q-Analog Klein Bottle Lie Algebra

2014 ◽  
Vol 21 (04) ◽  
pp. 561-574
Author(s):  
Cuipo Jiang ◽  
Jingjing Jiang ◽  
Yufeng Pei
Keyword(s):  
Lie Algebra ◽  
Bilinear Form ◽  
Central Extension ◽  
Klein Bottle ◽  
Universal Central Extension ◽  
Finitely Generated ◽  
Invariant Bilinear Form ◽  
Infinite Dimensional ◽  
Simple Lie Algebra ◽  
Infinite Dimensional Lie Algebra

In this paper, we study an infinite-dimensional Lie algebra ℬq, called the q-analog Klein bottle Lie algebra. We show that ℬq is a finitely generated simple Lie algebra with a unique (up to scalars) symmetric invariant bilinear form. The derivation algebra and the universal central extension of ℬq are also determined.

ON ISOMORPHISM CRITERIA FOR LEIBNIZ CENTRAL EXTENSIONS OF A LINEAR DEFORMATION OF μn

2011 ◽  
Vol 21 (05) ◽  
pp. 715-729 ◽  
Author(s):  
ISAMIDDIN S. RAKHIMOV ◽  
MUNTHER A. HASSAN
Keyword(s):  
Lie Algebra ◽  
Classification Problems ◽  
Linear Deformation ◽  
Central Extensions ◽  
Leibniz Algebras ◽  
Filiform Lie Algebra

This paper deals with the classification problems of Leibniz central extensions of linear deformations of a Lie algebra. It is known that any n-dimensional filiform Lie algebra can be represented as a linear deformation of n-dimensional filiform Lie algebra μn given by the brackets [ei, e0] = ei+1, i = 0,1,…,n - 2, in a basis {e0, e1,…,en - 1}. In this paper we consider a linear deformation of μn and its Leibniz central extensions. The resulting algebras are Leibniz algebras, this class is denoted here by Ced (μn). We choose an appropriate basis of Ced (μn) and give general isomorphism criteria. By using the isomorphism criteria, one can classify the class Ced (μn) for any fixed n. Two relevant maple programs are provided.

scholarly journals The second cohomology group of simple Leibniz algebras

2018 ◽  
Vol 17 (12) ◽  
pp. 1850222 ◽  
Author(s):  
J. Q. Adashev ◽  
M. Ladra ◽  
B. A. Omirov
Keyword(s):  
Lie Algebra ◽  
Cohomology Group ◽  
Leibniz Algebra ◽  
Leibniz Algebras ◽  
Leibniz Cohomology

In this paper, we prove some general results on Leibniz 2-cocyles for simple Leibniz algebras. Applying these results, we prove the triviality of the second Leibniz cohomology for a simple Leibniz algebra with coefficients in itself, whose associated Lie algebra is isomorphic to [Formula: see text].

Ganea Term for Homology of Leibniz n-Algebras

2005 ◽  
Vol 12 (04) ◽  
pp. 629-634 ◽  
Author(s):  
J. M. Casas
Keyword(s):  
Exact Sequence ◽  
Central Extension ◽  
Central Extensions ◽  
Leibniz Algebras

We extend the five-term exact sequence of homology with trivial coefficients of Leibniz n-algebras [Formula: see text] associated to a central extension of Leibniz n-algebras [Formula: see text] by means of a sixth term which is a generalization of the Ganea term for homology of Leibniz algebras. We use this sequence in order to analyze several questions related with the centre and central extensions of a Leibniz n-algebra.

On isomorphism criterion for a subclass of complex filiform Leibniz algebras

2017 ◽  
Vol 27 (07) ◽  
pp. 953-972
Author(s):  
I. S. Rakhimov ◽  
A. Kh. Khudoyberdiyev ◽  
B. A. Omirov ◽  
K. A. Mohd Atan
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Central Extension ◽  
Linear Deformation ◽  
Leibniz Algebras ◽  
Structure Constants ◽  
Filiform Lie Algebras ◽  
Filiform Lie Algebra ◽  
Isomorphism Criterion

In this paper, we present an algorithm to give the isomorphism criterion for a subclass of complex filiform Leibniz algebras arising from naturally graded filiform Lie algebras. This subclass appeared as a Leibniz central extension of a linear deformation of filiform Lie algebra. We give the table of multiplication choosing appropriate adapted basis, identify the elementary base changes and describe the behavior of structure constants under these base changes, then combining them the isomorphism criterion is given. The final result of calculations for one particular case also is provided.

scholarly journals Structure of the Extended Schrödinger–Virasoro Lie Algebra $\widetilde{\mathfrak{sv}}$

2009 ◽  
Vol 16 (04) ◽  
pp. 549-566 ◽  
Author(s):  
Shoulan Gao ◽  
Cuipo Jiang ◽  
Yufeng Pei
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Statistical Physics ◽  
Two Dimensional ◽  
Universal Central Extension ◽  
Central Extensions ◽  
Infinite Dimensional ◽  
Conformal Field

We study the derivations, the central extensions and the automorphism group of the extended Schrödinger–Virasoro Lie algebra [Formula: see text], introduced by Unterberger in the context of two-dimensional conformal field theory and statistical physics. Moreover, we show that [Formula: see text] is an infinite-dimensional complete Lie algebra, and the universal central extension of [Formula: see text] in the category of Leibniz algebras is the same as that in the category of Lie algebras.

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