Symplectic, Product and Complex Structures on 3‐Lie Algebras

Sheng, Yunhe; Tang, Rong

doi:10.1002/9781119818175.ch8

Cited by 9 publications

(13 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By the definitions of a 3-pre-Lie algebra and a representation of a 3-Lie algebra, we immediately obtain Definition 2.14. [20] A representation of a 3-pre-Lie algebra (A, {•, •, •}) on a vector space V consists of a pair (l, r), where l : ∧ 2 A → l(V) is a representation of the 3-Lie algebra A c on V and r : ⊗ 2 A → l(V) is a linear map such that for all x 1 , x 2 , x 3 , x 4 ∈ A, the following equalities hold:…”

Section: Preliminaries and Basicsmentioning

confidence: 99%

“…Proposition 2.15. [20] Let (A, {•, •, •}) be a 3-pre-Lie algebra, V a vector space and l, r : ⊗ 2 A → l(V) two linear maps. Then (V, l, r) is a representation of A if and only if there is a 3-pre-Lie algebra structure (called the semi-direct product) on the direct sum A ⊕ V of vector spaces, defined by…”

Section: Preliminaries and Basicsmentioning

confidence: 99%

“…Proposition 2.16. [20] Let (V, l, r) be a representation of a 3-pre-Lie algebra (A, {•, •, •}). Then l − rτ + r is a representation of the sub-adjacent 3-Lie algebra (A c , [•, •, •] C ) on the vector space V.…”

Section: Preliminaries and Basicsmentioning

confidence: 99%

“…Proposition 2.17. [20] Let (l, r) be a representation of a 3-pre-Lie algebra (A, {•, •, •}) on a vector space V. Then (l * − r * τ + r * , −r * ) is a representation of the 3-pre-Lie algebra (A, {•, •, •}) on the vector space V * , which is called the dual representation of the representation (V, l, r).…”

Section: Preliminaries and Basicsmentioning

confidence: 99%

See 3 more Smart Citations

3-L-dendriform algebras and generalized derivations

Chtioui

Mabrouk

2021

Filomat

View full text Add to dashboard Cite

The main goal of this paper is to introduce the notion of 3-L-dendriform algebras which are the dendriform version of 3-pre-Lie algebras. In fact they are the algebraic structures behind the O-operator of 3-pre-Lie algebras. They can be also regarded as the ternary analogous of L-dendriform algebras. Moreover, we study the generalized derivations of 3-L-dendriform algebras. Finally, we explore the spaces of quasi-derivations, the centroids and the quasi-centroids and give some properties.

show abstract

Section: Preliminaries and Basicsmentioning

confidence: 99%

Section: Preliminaries and Basicsmentioning

confidence: 99%

Section: Preliminaries and Basicsmentioning

confidence: 99%

Section: Preliminaries and Basicsmentioning

confidence: 99%

See 2 more Smart Citations

3-L-dendriform algebras and generalized derivations

Chtioui

Mabrouk

2021

Filomat

View full text Add to dashboard Cite

show abstract

“…These Lie algebras are known by different names such as Filippov algebras, them in [18]. Later, the new notions such as 3-Lie color algebra, 3-Hom-Lie color algebra and 3-pre-Lie algebra were constructed and studied in [5,20,25,30].…”

Section: Introductionmentioning

confidence: 99%

Quadratic and symplectic structures on 3-(Hom)-$ρ$-Lie algebras

Bagheri,

Peyghan

2021

Preprint

View full text Add to dashboard Cite

Our purpose in this paper is the generalization of the notions of quadratic and symplectic structures to the case of 3-(Hom)-ρ-Lie algebras. We describe some properties of them by expressing the related lemmas and theorems. Also, we introduce the concept of 3-pre-(Hom)-ρ-Lie algebras and define their representation.Nambu-Lie algebras, Lie n-algebras. The special case of these algebras are 3-Lie algebras and in the Hom case 3-Hom-Lie algebras, which extracted from 3-Lie algebras base on Hom-Lie algebras. These algebras were defined and checked by H. Ataguema, A. Makhlouf and S. Silvestrov in [2]. Y. Liu, L. Chena and Y. Ma took into consideration these algebras and studied the representations and module-extensions on

show abstract

3-Lie-Rinehart Algebras

Bai¹,

Li²,

Wu³

2019

Preprint

View full text Add to dashboard Cite

In this paper, we define a class of 3-algebras which are called 3-Lie-Rinehart algebras. A 3-Lie-Rinehart algebra is a triple (L, A, ρ), where A is a commutative associative algebra, L is an A-module, (A, ρ) is a 3-Lie algebra L-module and ρ(L, L) ⊆ Der(A). We discuss the basic structures, actions and crossed modules of 3-Lie-Rinehart algebras and construct 3-Lie-Rinehart algebras from given algebras, we also study the derivations from 3-Lie-Rinehart algebras to 3-Lie A-algebras. From the study, we see that there is much difference between 3-Lie algebras and 3-Lie-Rinehart algebras.

show abstract

Symplectic, Product and Complex Structures on 3‐Lie Algebras

Cited by 9 publications

References 27 publications

3-L-dendriform algebras and generalized derivations

3-L-dendriform algebras and generalized derivations

Quadratic and symplectic structures on 3-(Hom)-$ρ$-Lie algebras

3-Lie-Rinehart Algebras

Contact Info

Product

Resources

About