Generalized Derivations of n-BiHom-Lie algebras

Abdeljelil, Amine Ben; Elhamdadi, Mohamed; Kaygorodov, Ivan; Makhlouf, Abdenacer

doi:10.48550/arxiv.1901.09750

Cited by 5 publications

(5 citation statements)

References 29 publications

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“…In this section, we introduce the notion of (α s , β r )-derivations of 3-BiHom-Lie superalgebras generalize the notion of (α s , β r )-derivations of 3-BiHom-Lie algebras introduced in [2] and we give some results.…”

Section: Definitions and Notationsmentioning

confidence: 99%

3-BiHom-Lie superalgebras induced by BiHom-Lie superalgebras

Hassine

Mabrouk

Ncib

2020

Linear and Multilinear Algebra

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The purpose of this paper is to study the relationships between a BiHom-Lie superalgebras and its induced 3-BiHom-Lie superalgebras. We introduce the notion of (α s , β r )derivation, (α s , β r )-quasiderivation and generalized (α s , β r )-derivation of 3-BiHom-Lie superalgebras, and their relation with derivation of BiHom-Lie superalgebras. We introduce also the concepts of Rota-Baxter operators and Nijenhuis Operators of BiHom 3-Lie superalgebras. We also explore the construction of 3-BiHom-Lie superalgebras by using Rota-Baxter of BiHom-Lie superalgebras. also been studied in connection with many areas of mathematics and physics, including combinatorics, number theory, operators and quantum field theory [16]. Furthermore, Rota-Baxter operators on a Lie algebra are an operator form of the classical Yang-Baxter equations and contribute to the study of integrable systems [5]. Further Rota-Baxter 3-Lie algebras are closely related to pre-Lie algebras [6]. Rota-Baxter of multiplicative 3-ary Hom-Nambu-Lie algebras were introduced by Sun and Chen, in [14].Deformations of n-Lie algebras have been studied from several aspects. See [1,9] for more details. In particular, a notion of a Nijenhuis operator on a 3-Lie algebra was introduced in [15] in the study of the 1-order deformations of a 3-Lie algebra. But there are some quite strong conditions in this definition of a Nijenhuis operator. In the case of Lie algebras, one could obtain fruitful results by considering one-parameter infinitesimal deformations, i.e. 1-order deformations. However, for n-Lie algebras, we believe that one should consider (n − 1)-order deformations to obtain similar results. In [9], for 3-Lie algebras, the author had already considered 2-order deformations. For the case of Hom-Lie superalgebras, the authors in [13] give the notion of Hom-Nijenhuis operator.Thus it is time to study 3-BiHom-Lie superalgebras, Rota-Baxter algebras and Nijenhuis operator together to get a suitable definition of Rota-Baxter of 3-BiHom-Lie superalgebras induced by BiHom-Lie superalgebras. Similarly, we give the relationship between Nijenhuis operator of 3-BiHom-Lie superalgebras and BiHom-Lie superalgebras. This paper is organized as follows: In Section 1, we recall the concepts of BiHom-Lie superalgebras and introduce the notion of 3-BiHom-Lie superalgebras. The construction of 3-BiHom-Lie superalgebras induced by BiHom-Lie superalgebras are established in Section 2. In section 3, we give the definition of (α s , β r )-derivation and (α s , β r )quasiderivation of 3-BiHom-Lie superalgebras. In section 4, we give the definition of Rota-Baxter of 3-BiHom-Lie superalgebras and the realizations of Rota-Baxter of 3-BiHom-Lie superalgebras from Rota-Baxter BiHom-Lie superalgebras. The Section 5 is dedicated to study the second order deformation of 3-BiHom-Lie superalgebras, and introduce the notion of Nijenhuis operator on 3-BiHom-Lie superalgebras, which could generate a trivial deformation. In the other part of this section we give some properties and results of Nijen...

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Section: Definitions and Notationsmentioning

confidence: 99%

3-BiHom-Lie superalgebras induced by BiHom-Lie superalgebras

Hassine

Mabrouk

Ncib

2020

Linear and Multilinear Algebra

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“…Since the pioneering works [18,[29][30][31][32]47], Hom-algebra structures have developed in a popular broad area with increasing number of publications in various directions. Hom-algebra structures include their classical counterparts and open new broad possibilities for deformations, extensions to Hom-algebra structures of representations, homology, cohomology and formal deformations, Hommodules and hom-bimodules, Hom-Lie admissible Hom-coalgebras, Hom-coalgebras, Hom-bialgebras, Hom-Hopf algebras, L-modules, L-comodules and Hom-Lie quasibialgebras, n-ary generalizations of biHom-Lie algebras and biHom-associative algebras and generalized derivations, Rota-Baxter operators, Hom-dendriform color algebras, Rota-Baxter bisystems and covariant bialgebras, Rota-Baxter cosystems, coquasitriangular mixed bialgebras, coassociative Yang-Baxter pairs, coassociative Yang-Baxter equation and generalizations of Rota-Baxter systems and algebras, curved Ooperator systems and their connections with tridendriform systems and pre-Lie algebras, BiHom-algebras, BiHom-Frobenius algebras and double constructions, infinitesimal biHom-bialgebras and Hom-dendriform D-bialgebras, and category theory of Homalgebras [2,3,[5][6][7][8][9][10][11][12]15,16,[19][20][21][22][23][24]29,[32][33][34][37][38][39][40]42,45,[48][49][50]…”

Section: Introductionmentioning

confidence: 99%

Hom-pre-Malcev and Hom-M-Dendriform algebras

Harrathi¹,

Mabrouk²,

Ncib³

et al. 2021

Preprint

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The purpose of this paper is to introduce and study the notion of Hom-pre-Malcev algebra which may be seen as a Malcev algebra whose product can be decomposed into two compatible pieces. We show also that bimodules over Malcev and pre-Malcev are twisted into bimodules over Hom-Malcev and Hom-pre-Malcev via endomorphisms respectively. Furthermore, we introduce the notion of Hom-M-dendriform algebra and show the connections between all these algebraic structures using O-operators.

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“…In particular, the definition of n-Bihom-Lie algebras and n-Bihom-Associative algebras were introduced in [11]. Then the generalized derivations of 3-BiHom-Lie superalgebras and n-BiHom-Lie algebras are studied in [5,6].…”

Section: Introductionmentioning

confidence: 99%

The construction of 3-Bihom-Lie algebras

Chen

2020

Preprint

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The purpose of this paper is to study the construction of 3-Bihom-Lie algebras. We give some ways of constructing 3-Bihom-Lie algebras from 3-Bihom-Lie algebras and 3totally Bihom-associative algebras. Furthermore, we introduce T θ -extensions and T * θextensions of 3-Bihom-Lie algebras and prove the necessary and sufficient conditions for a 2n-dimensional quadratic 3-Bihom-Lie algebra to be isomorphic to a T * θ -extension.

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Generalized Derivations of n-BiHom-Lie algebras

Cited by 5 publications

References 29 publications

3-BiHom-Lie superalgebras induced by BiHom-Lie superalgebras

3-BiHom-Lie superalgebras induced by BiHom-Lie superalgebras

Hom-pre-Malcev and Hom-M-Dendriform algebras

The construction of 3-Bihom-Lie algebras

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