Transposed Hom-Poisson and Hom-pre-Lie Poisson algebras and bialgebras

Abstract

The notions of transposed Hom-Poisson and Hom-pre-Lie Poisson algebras are introduced. Their bimodules and matched pairs are defined and the relevant properties and theorems are given. The notion of Manin triple of transposed Hom-Poisson algebras is introduced, and its equivalence to the transposed Hom-Poisson bialgebras is investigated. The notion of O-operator is exploited to illustrate the relations existing between transposed Hom-Poisson and Hom-pre-Lie Poisson algebras.

On Hom-Nambu-Poisson superalgebras structures and their representations

“…which is gives by many authors (see [6,22,28,32,40]). More applications of (Hom)-Lie superalgebras, (Hom)-Lie color algebras and some results on the structures of (Hom)-Poisson superalgebras have been widely investigated [5,2,4,23,12,36,20,13].…”

Transposed BiHom-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.…”

“…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.…”

“…Transposed Poisson algebras were presented in [1] by exchanging the operations • and [, ] in the compatible condition of Poisson algebras, and at the same time a factor 2 appears on the left-hand side. It was also studied in [8,16,32] recently. A 3-Lie algebra is a vector space A endowed with a ternary skew-symmetric operation satisfying the ternary Jacobi identity [2,4,9].…”

“…In Section 2, we introduce the notion of TBP algebras (Definition 2.1) which extends transposed Poisson algebra defined in [1]. We note here when α = β, it is still different from the transposed Hom-Poisson algebra introduced in [16]. A TBP algebra can be constructed by a commutative associative algebra with a derivation (Proposition 2.4).…”