scholarly journals A New Approach to Representations of 3-Lie Algebras and Abelian Extensions

2017 ◽  
Vol 20 (6) ◽  
pp. 1415-1431 ◽  
Author(s):  
Jiefeng Liu ◽  
Abdenacer Makhlouf ◽  
Yunhe Sheng
Keyword(s):  
Lie Algebras ◽  
New Approach ◽  
Abelian Extensions

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

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.

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 Nijenhuis operators on pre-Lie algebras

2019 ◽  
Vol 21 (07) ◽  
pp. 1850050 ◽  
Author(s):  
Qi Wang ◽  
Yunhe Sheng ◽  
Chengming Bai ◽  
Jiefeng Liu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Regular Representation ◽  
Complex Structure ◽  
Geometric Interpretation ◽  
Dual Representation ◽  
New Approach ◽  
Lie Bracket ◽  
Graded Lie Algebra ◽  
Dendriform Algebras

First we use a new approach to define a graded Lie algebra whose Maurer–Cartan elements characterize pre-Lie algebra structures. Then using this graded Lie bracket, we define the notion of a Nijenhuis operator on a pre-Lie algebra which generates a trivial deformation of this pre-Lie algebra. There are close relationships between [Formula: see text]-operators, Rota–Baxter operators and Nijenhuis operators on a pre-Lie algebra. In particular, a Nijenhuis operator “connects” two [Formula: see text]-operators on a pre-Lie algebra whose any linear combination is still an [Formula: see text]-operator in certain sense and hence compatible [Formula: see text]-dendriform algebras appear naturally as the induced algebraic structures. For the case of the dual representation of the regular representation of a pre-Lie algebra, there is a geometric interpretation by introducing the notion of a pseudo-Hessian–Nijenhuis structure which gives rise to a sequence of pseudo-Hessian and pseudo-Hessian–Nijenhuis structures. Another application of Nijenhuis operators on pre-Lie algebras in geometry is illustrated by introducing the notion of a para-complex structure on a pre-Lie algebra and then studying para-complex quadratic pre-Lie algebras and para-complex pseudo-Hessian pre-Lie algebras in detail. Finally, we give some examples of Nijenhuis operators on pre-Lie algebras.

scholarly journals A new approach for finding standard heat equation and a special Newell-whitehead equation

2019 ◽  
Vol 23 (3 Part A) ◽  
pp. 1629-1636
Author(s):  
Xiu-Rong Guo ◽  
Yu-Feng Zhang ◽  
Mei Guo ◽  
Zheng-Tao Liu
Keyword(s):  
Heat Equation ◽  
Lie Algebras ◽  
Integrable Model ◽  
Block Matrix ◽  
Integrable Hierarchy ◽  
New Approach ◽  
Akns Hierarchy ◽  
Standard Heat ◽  
Integrable Couplings ◽  
Isospectral Problem

Under a frame of 2 ? 2 matrix Lie algebras, Tu and Meng [9] once established a united integrable model of the Ablowitz-Kaup-Newel-Segur (AKNS) hierarchy, the D-AKNS hierarchy, the Levi hierarchy and the TD hierarchy. Based on this idea, we introduce two block-matrix Lie algebras to present an isospectral problem, whose compatibility condition gives rise to a type of integrable hierarchy which can be reduced to the Levi hierarchy and the AKNS hierarchy, and so on. A united integrable model obtained by us in the paper is different from that given by Tu and Meng. Specially, the main result in the paper can be reduced to two new various integrable couplings of the Levi hierarchy, from which we again obtain the standard heat equation and a special Newell-Whitehead equation.

Multiplicity-Free Gradings on Semisimple Lie and Jordan Algebras and Skew Root Systems

2019 ◽  
Vol 26 (01) ◽  
pp. 123-138
Author(s):  
Gang Han ◽  
Yucheng Liu ◽  
Kang Lu
Keyword(s):  
Abelian Group ◽  
Lie Algebras ◽  
Root Systems ◽  
Jordan Algebras ◽  
Homogeneous Component ◽  
New Approach ◽  
Jordan Type ◽  
Semisimple Lie Algebras ◽  
Group Gradings ◽  
Multiplicity Free

A G-grading on an algebra, where G is an abelian group, is called multiplicity-free if each homogeneous component of the grading is 1-dimensional. We introduce skew root systems of Lie type and skew root systems of Jordan type, and use them to construct multiplicity-free gradings on semisimple Lie algebras and on semisimple Jordan algebras respectively. Under certain conditions the corresponding Lie (resp., Jordan) algebras are simple. Two families of skew root systems of Lie type (resp., of Jordan type) are constructed and the corresponding Lie (resp., Jordan) algebras are identified. This is a new approach to study abelian group gradings on Lie and Jordan algebras.

The Wells map for abelian extensions of 3-Lie algebras

2019 ◽  
Vol 69 (4) ◽  
pp. 1133-1164
Author(s):  
Youjun Tan ◽  
Senrong Xu
Keyword(s):  
Lie Algebras ◽  
Abelian Extensions
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