scholarly journals Topics on n-ary Algebraic Structures

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.

scholarly journals Generalized Reynolds operators on 3-Lie algebras and NS-3-Lie algebras

2021 ◽  
Author(s):  
Shuai Hou ◽  
Yunhe Sheng
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Group ◽  
Algebra Structure ◽  
Algebraic Structures ◽  
Reynolds Operator ◽  
Infinitesimal Deformations ◽  
Special Case

In this paper, first, we introduce the notion of a generalized Reynolds operator on a [Formula: see text]-Lie algebra [Formula: see text] with a representation on [Formula: see text]. We show that a generalized Reynolds operator induces a 3-Lie algebra structure on [Formula: see text], which represents on [Formula: see text]. By this fact, we define the cohomology of a generalized Reynolds operator and study infinitesimal deformations of a generalized Reynolds operator using the second cohomology group. Then we introduce the notion of an NS-[Formula: see text]-Lie algebra, which produces a 3-Lie algebra with a representation on itself. We show that a generalized Reynolds operator induces an NS-[Formula: see text]-Lie algebra naturally. Thus NS-[Formula: see text]-Lie algebras can be viewed as the underlying algebraic structures of generalized Reynolds operators on [Formula: see text]-Lie algebras. Finally, we show that a Nijenhuis operator on a 3-Lie algebra gives rise to a representation of the deformed 3-Lie algebra and a 2-cocycle. Consequently, the identity map will be a generalized Reynolds operator on the deformed 3-Lie algebra. We also introduce the notion of a Reynolds operator on a [Formula: see text]-Lie algebra, which can serve as a special case of generalized Reynolds operators on 3-Lie algebras.

scholarly journals Representations of Bihom-Lie Algebras

2022 ◽  
Vol 29 (01) ◽  
pp. 125-142
Author(s):  
Yongsheng Cheng ◽  
Huange Qi
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Adjoint Representation ◽  
Trivial Representation ◽  
Central Extensions ◽  
Linear Maps

A Bihom-Lie algebra is a generalized Hom-Lie algebra endowed with two commuting multiplicative linear maps. In this paper, we study representations of Bihom-Lie algebras. In particular, derivations, central extensions, derivation extensions, the trivial representation and the adjoint representation of Bihom-Lie algebras are studied in detail.

scholarly journals Invariant metrics on central extensions of quadratic Lie algebras

2019 ◽  
Vol 19 (12) ◽  
pp. 2050224
Author(s):  
R. García-Delgado ◽  
G. Salgado ◽  
O. A. Sánchez-Valenzuela
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
General Structure ◽  
Invariant Metrics ◽  
Invariant Metric ◽  
Central Extensions ◽  
Direct Sum ◽  
Quadratic Lie Algebra

A quadratic Lie algebra is a Lie algebra endowed with a symmetric, invariant and nondegenerate bilinear form; such a bilinear form is called an invariant metric. The aim of this work is to describe the general structure of those central extensions of quadratic Lie algebras which in turn have invariant metrics. The structure is such that the central extensions can be described algebraically in terms of the original quadratic Lie algebra, and geometrically in terms of the direct sum decompositions that the invariant metrics involved give rise to.

Some Properties of the c-Nilpotent Multiplier and c-Covers of Lie Algebras

2014 ◽  
Vol 21 (03) ◽  
pp. 421-426 ◽  
Author(s):  
Mohammad Reza Rismanchian ◽  
Mehdi Araskhan
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Central Extensions ◽  
Finite Dimensional

This paper is devoted to present some properties of the c-nilpotent multiplier [Formula: see text] and some features of c-central extensions of a (finite dimensional) Lie algebra L. Moreover, we give the structure of all c-covers of Lie algebras whose c-nilpotent multipliers have the Hopfian property.

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.

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.

Dual Space Derivations and H2(L, F) of Modular Lie Algebras

1987 ◽  
Vol 39 (5) ◽  
pp. 1078-1106 ◽  
Author(s):  
Rolf Farnsteiner
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Group ◽  
Dual Space ◽  
Positive Characteristic ◽  
Base Field ◽  
Central Extensions ◽  
Simple Lie Algebras ◽  
Early Results ◽  
Modular Lie Algebras

It is well-known that the classical vanishing results of the cohomology theory of Lie algebras depend on the characteristic of the underlying base field. The theorems of Cartan and Zassenhaus, for instance, entail that non-modular simple Lie algebras do not admit non-trivial central extensions. In contrast, early results by Block [3] prove that this conclusion loses its validity if the underlying base field has positive characteristic.Central extensions of a given Lie algebra L, or equivalently its second cohomology group H(L, F), can be conveniently described by means of derivations φ:L → L*.

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 EXTENSIONS OF (SUPER) LIE ALGEBRAS

2009 ◽  
Vol 11 (05) ◽  
pp. 709-737 ◽  
Author(s):  
ALICE FIALOWSKI ◽  
MICHAEL PENKAVA
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Extension Problem ◽  
Infinitesimal Deformations ◽  
Cohomological Interpretation

In this paper, we give a purely cohomological interpretation of the extension problem for (super) Lie algebras, that is, the problem of extending a Lie algebra by another Lie algebra. We then give a similar interpretation of infinitesimal deformations of extensions. In particular, we consider infinitesimal deformations of representations of a Lie 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.

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