scholarly journals Cohomologies of n-Lie Algebras with Derivations

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2452
Author(s):  
Qinxiu Sun ◽  
Zhixiang Wu
Keyword(s):  
Lie Algebras ◽  
Cohomology Groups ◽  
Trivial Representation ◽  
Central Extensions

The goal of this paper is to study cohomological theory of n-Lie algebras with derivations. We define the representation of an n-LieDer pair and consider its cohomology. Likewise, we verify that a cohomology of an n-LieDer pair could be derived from the cohomology of associated LeibDer pair. Furthermore, we discuss the (n−1)-order deformations and the Nijenhuis operator of n-LieDer pairs. The central extensions of n-LieDer pairs are also investigated in terms of the first cohomology groups with coefficients in the trivial representation.

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 Integrating central extensions of Lie algebras via Lie 2-groups

2016 ◽  
Vol 18 (6) ◽  
pp. 1273-1320 ◽  
Author(s):  
Christoph Wockel ◽  
Chenchang Zhu
Keyword(s):  
Lie Algebras ◽  
Central Extensions

scholarly journals Central extensions of the quasi-orthogonal Lie algebras

1998 ◽  
Vol 31 (5) ◽  
pp. 1373-1394 ◽  
Author(s):  
J A de Azcárraga ◽  
F J Herranz ◽  
J C Pérez Bueno ◽  
M Santander
Keyword(s):  
Lie Algebras ◽  
Central Extensions

Central extensions of Lie algebras graded by finite root systems

2000 ◽  
Vol 316 (3) ◽  
pp. 499-527 ◽  
Author(s):  
Bruce Allison ◽  
Georgia Benkart ◽  
Yun Gao
Keyword(s):  
Lie Algebras ◽  
Root Systems ◽  
Central Extensions

Representations and cohomologies of regular Hom-pre-Lie algebras

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.

Central Extensions and Derivations of Generalized Schrödinger-Virasoro Algebras

2012 ◽  
Vol 19 (04) ◽  
pp. 735-744 ◽  
Author(s):  
Wei Wang ◽  
Junbo Li ◽  
Bin Xin
Keyword(s):  
Lie Algebras ◽  
Natural Generalization ◽  
Additive Subgroup ◽  
Central Extensions ◽  
Infinite Dimensional

Let 𝔽 be a field of characteristic 0, G an additive subgroup of 𝔽, s ∈ 𝔽 such that s ∉ G and 2s ∈ G. A class of infinite-dimensional Lie algebras [Formula: see text] called generalized Schrödinger-Virasoro algebras was defined by Tan and Zhang, which is a natural generalization of Schrödinger-Virasoro algebras. In this paper, central extensions and derivations of [Formula: see text] are determined.

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