Constructing Hom-Poisson color algebras

Abstract

We give some constructions of Hom-Poisson color algebras first from a Hom-associative color algebra which twisting map is an averaging operator, then from a given Hom-Poisson color algebra and an averaging operator and finally from a Hom-post-Poisson color algebra. Then we show that any Hom-pre-Poisson color algebra leads to a Hom-Poisson color algebra. Conversly, we prove that any Hom-Poisson color algebra turn to a Hom-pre-Poisson color algebra via Rota-Baxter operator.Koszul dualization. Some examples and related structures are given. Among other results, the authors proved in [5] that tridendriform and Rota-Baxter Lie algebras lead to the structures of post-Lie algebras. Post-Poisson algebras [6] are algebraic structures containing post-Lie algebras [4], [9] and pre-Poisson algebras [13] which contain pre-Lie algebras (or left-symmetric algebras) and Zinbiel algebras, both structures being connected by some compatibility conditions.Classical (or ordinary) algebras were extended to generalized (or color or graded) algebras, among which one can cite : Simple Jordan color algebras arising from associative graded algebras [11], Representations and cocycle twists of color Lie algebras [16], On the classification of 3-dimensional coloured Lie algebras [15]. These structures are well-known to physicists and to mathematicans studying differential geometry and homotopy theory. They were extended to the Hom-setting by studying Hom-Lie superalgebras, Rota-Baxter operator on pre-Lie superalgebras and beyound [8] and Hom-Lie color algebras [12]. Color Hom-Poisson algebras were introduced in [10] as generalization of Hom-Poisson algebras [1].The aim of this paper is to introduce and study the relationship among commutative Hom-tridendriform color algebras and Hom-post-Poisson color algebras. The paper is organized as follows. In section 2, we recall definition of Hom-Poisson color algebras and define averaging operator on Hom-associative and Hom-Lie color algebras. We introduce Hom-post-Lie color algebras, and give some procedures of construction of Hom-post-Lie color algebras form either a given one or from Hom-Lie Rota-Baxter color algebras. The section 3 is devoted to the main results of this paper i.e. some constructions of Hom-Poisson color algebras from other algebraic structures.Throughout this paper, all graded vector spaces are assumed to be over a field K of characteristic different from 2. PreliminariesWe give basic notions and give some of their properties.Let G be an abelian group. A vector space V is said to be a G-graded if, there exists a family (V a ) a∈G of vector subspaces of V such thatAn element x ∈ V is said to be homogeneous of degree a ∈ G if x ∈ V a . We denote H(V ) the set of all homogeneous elements in V .Let V = ⊕ a∈G V a and V = ⊕ a∈G V a be two G-graded vector spaces. A linear mapping f : V → V is said to be homogeneous of degree b if f (V a ) ⊆ V a+b , ∀a ∈ G.

Simply Complete Hom-Lie Superalgebras and Decomposition of Complete Hom-Lie Superalgebras

“…The Hom-Lie superalgebras and the more general color quasi-Lie algebras provide new general parametric families of non-associative structures, extending and interpolating on the fundamental level of defining identities between the Lie algebras, Lie superalgebras, color Lie algebras and some other important related non-associative structures, their deformations and discretizations, in the special interesting ways which may be useful for unification of models of classical and quantum physics, geometry and symmetry analysis, and also in algebraic analysis of computational methods and algorithms involving linear and non-linear discretizations of differential and integral calculi. Investigation of color hom-Lie algebras and hom-Lie superalgebras and n-ary generalizations have been further expanded recently in [1,2,7,8,[11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]33,42,43,[48][49][50]60,61,64,65,[69][70][71][72][73]75].…”

Some structures of Hom-Poisson color algebras

“…However, Hom-Poisson color algebras are introduced in [3] as the colored version verion of Hom-Poisson algebras introduced in [2]. The authers in [2] give some constructions of Hom-Poisson color algebras from Hom-associative color algebras which twisting map is an averaging operator or from a given Hom-Poisson color algebra together with an averaging operator or from a Hom-post-Poisson color algebra. In particular, they show that any Hom-pre-Poisson color algebra leads to a Hom-Poisson color algebra.…”

“…The goal of this paper is to give a contuation of constructions of Hom-Poisson color algebras [2]. While many authers working on Hom-algebras use a morphism of Hom-algebras to builds another one, we ask our self if there are other kind of twist which are not morphism such that we can get Hom-algebraic structures from others one.…”