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

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.

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On isomorphism criterion for a subclass of complex filiform Leibniz algebras

International Journal of Algebra and Computation ◽

10.1142/s0218196717500448 ◽

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.

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Lie triple systems and Leibniz algebras

Georgian Mathematical Journal ◽

10.1515/gmj-2020-2053 ◽

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.

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ON ONE-DIMENSIONAL LEIBNIZ CENTRAL EXTENSIONS OF A FILIFORM LIE ALGEBRA

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972711002371 ◽

2011 ◽

Vol 84 (2) ◽

pp. 205-224 ◽

Cited By ~ 15

Author(s):

ISAMIDDIN S. RAKHIMOV ◽

MUNTHER A. HASSAN

Keyword(s):

Lie Algebra ◽

Multiplication Table ◽

Central Extensions ◽

Invariant Functions ◽

One Dimensional ◽

Isomorphism Classes ◽

Parametric Families ◽

Low Dimensional ◽

Filiform Lie Algebra

AbstractThe paper deals with the classification of Leibniz central extensions of a filiform Lie algebra. We choose a basis with respect to which the multiplication table has a simple form. In low-dimensional cases isomorphism classes of the central extensions are given. In the case of parametric families of orbits, invariant functions (orbit functions) are provided.

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On universal central extensions of Hom-Leibniz algebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498814500534 ◽

2014 ◽

Vol 13 (08) ◽

pp. 1450053 ◽

Cited By ~ 3

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.

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Topics on n-ary Algebraic Structures

Acta Polytechnica ◽

10.14311/1179 ◽

2010 ◽

Vol 50 (3) ◽

Author(s):

J. A. de Azcárraga ◽

J. M. Izquierdo

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Algebraic Structures ◽

Central Extensions ◽

Generalized Poisson ◽

Leibniz Algebras ◽

Infinitesimal Deformations ◽

Generalized Lie Algebras

We review the basic definitions and properties of two types of n-ary structures, the Generalized Lie Algebras (GLA) and the Filippov (≡ n-Lie) algebras (FA), as well as those of their Poisson counterparts, the Generalized Poisson (GPS) and Nambu-Poisson (N-P) structures. We describe the Filippov algebra cohomology complexes relevant for the central extensions and infinitesimal deformations of FAs. It is seen that semisimple FAs do not admit central extensions and, moreover, that they are rigid. This extends Whitehead’s lemma to all n ≥ 2, n = 2 being the original Lie algebra case. Some comments onn-Leibniz algebras are also made.

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

Asian-European Journal of Mathematics ◽

10.1142/s1793557122500541 ◽

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.

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Central Extensions of Lie Groups Preserving a Differential Form

International Mathematics Research Notices ◽

10.1093/imrn/rnaa085 ◽

2020 ◽

Author(s):

Tobias Diez ◽

Bas Janssens ◽

Karl-Hermann Neeb ◽

Cornelia Vizman

Keyword(s):

Lie Groups ◽

Lie Algebra ◽

Lie Group ◽

Differential Form ◽

Integral Form ◽

Symplectic Manifolds ◽

Central Extensions ◽

Mapping Space ◽

Canonical Class ◽

Differential Characters

Abstract Let $M$ be a manifold with a closed, integral $(k+1)$-form $\omega $, and let $G$ be a Fréchet–Lie group acting on $(M,\omega )$. As a generalization of the Kostant–Souriau extension for symplectic manifolds, we consider a canonical class of central extensions of ${\mathfrak{g}}$ by ${\mathbb{R}}$, indexed by $H^{k-1}(M,{\mathbb{R}})^*$. We show that the image of $H_{k-1}(M,{\mathbb{Z}})$ in $H^{k-1}(M,{\mathbb{R}})^*$ corresponds to a lattice of Lie algebra extensions that integrate to smooth central extensions of $G$ by the circle group ${\mathbb{T}}$. The idea is to represent a class in $H_{k-1}(M,{\mathbb{Z}})$ by a weighted submanifold $(S,\beta )$, where $\beta $ is a closed, integral form on $S$. We use transgression of differential characters from $ S$ and $ M $ to the mapping space $ C^\infty (S, M) $ and apply the Kostant–Souriau construction on $ C^\infty (S, M) $.

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Representations and cohomologies of regular Hom-pre-Lie algebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498820501492 ◽

2019 ◽

Vol 19 (08) ◽

pp. 2050149

Author(s):

Shanshan Liu ◽

Lina Song ◽

Rong Tang

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Cohomology Theory ◽

Trivial Representation ◽

Linear Deformation ◽

Dual Representations ◽

Regular Representations ◽

Equivalent Characterization ◽

Tensor Representations

In this paper, first we study dual representations and tensor representations of Hom-pre-Lie algebras. Then we develop the cohomology theory of regular Hom-pre-Lie algebras in terms of the cohomology theory of regular Hom-Lie algebras. As applications, we study linear deformations of regular Hom-pre-Lie algebras, which are characterized by the second cohomology groups of regular Hom-pre-Lie algebras with the coefficients in the regular representations. The notion of a Nijenhuis operator on a regular Hom-pre-Lie algebra is introduced which can generate a trivial linear deformation of a regular Hom-pre-Lie algebra. Finally, we introduce the notion of a Hessian structure on a regular Hom-pre-Lie algebra, which is a symmetric nondegenerate 2-cocycle with the coefficient in the trivial representation. We also introduce the notion of an [Formula: see text]-operator on a regular Hom-pre-Lie algebra, by which we give an equivalent characterization of a Hessian structure.

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