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

Abstract

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...

Transposed BiHom-Poisson algebras

“…Following [15], one can get 3-BiHom-Lie algebras which can be constructed from 3-Bihom-Lie algebras and 3-totally Bihom-associative algebras [17]. In [5], the authors studied the relationships between the BiHom-Lie superalgebras and its induced 3-BiHom-Lie superalgebras. In [16], the authors introduced the notion of transposed Hom-Poisson algebra and studied the bimodule and matched pair of transposed Hom-Poisson algebras.…”

“…Roughly speaking, a BiHom-associative algebra (or Lie algebra) is an algebra (or Lie algebra) such that the associativity (or Jacobi condition) is twisted by two (commuting) endomorphisms, for details see [10], which can be seen as an extension of Hom-type algebra [13] arising in quasi-deformations of Lie algebras of vector fields. Now there are so many research related to BiHom-type algebras, see refs [5,11,12,[15][16][17][18][19][20][21][23][24][25][26][27][28]. In [21], the authors introduced the notion of BiHom-Poisson algebras and gave a necessary and sufficient condition under which BiHom-Novikov-Poisson algebras (which are twisted generalizations of Novikov-Poisson algebras [30] and Hom-Novikov-Poisson algebras [31]) give rise to BiHom-Poisson algebras.…”

Bihom-pre-Lie superalgebras and related structures

“…Recall that a BiHom-algebra is an algebra in such a way that the identities defining the structure are twisted by two homomorphisms α and β. More applications of BiHom-Lie algebras, BiHom-algebras, BiHom-Lie superalgebras, BiHom-Lie admissible superalgebras and 3-BiHom-Lie superalgebras can be found in ( [18,23,46]).…”

Product and complex structures on 3-Bihom-Lie algebras

“…Considering n = 3, the authors given some constructions of 3-Bihom-Lie algebras in [10]. Moreover, in [6] the notion of Nijenhuis operators on 3-BiHom-Lie superalgebras were given, and the authors studied the relation between Nijenhuis operators and Rota-Baxter operators.…”