In this paper we introduce color Hom-Poisson algebras and show that every color Hom-associative algebra has a non-commutative Hom-Poisson algebra structure in which the Hom-Poisson bracket is the commutator bracket. Then we show that color Poisson algebras (respectively morphism of color Poisson algebras) turn to color Hom-Poisson algebras (respectively morphism of Color Hom-Poisson algebras) by twisting the color Poisson structure. Next we prove that modules over color Hom–associative algebras A extend to modules over the color Hom-Lie algebras L(A), where L(A) is the color Hom-Lie algebra associated to the color Hom-associative algebra A. Moreover, by twisting a color Hom-Poisson module structure map by a color Hom-Poisson algebra endomorphism, we get another one.

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Modules Over Color Hom-Poisson Algebras

The aim of this paper is to study some brackets defined on (; )-derivations satisfying quasi-Lie identities. Moreover, we provide examples of (p;q)-deformations of Witt and Virasoro algebras as well as sl(2) algebra. These constructions generalize the results obtained by Hartwig, Larsson and Silvestrov on -derivations, arising in connection with discretizations and deformations of algebras of vector fields.

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Brackets with (τ, σ )-derivations and (p, q)-deformations of Witt and Virasoro algebras

In this paper, we give some basic properties of the generalized derivation algebra ${\rm GDer}(L)$ of a Hom-Lie superalgebra $L$. In particular, we prove that ${\rm GDer}(L) = {\rm QDer}(L) + {\rm QC}(L)$, the sum of the quasiderivation algebra and the quasicentroid. We also prove that ${\rm QDer}(L)$ can be embedded as derivations in a larger Hom-Lie superalgebra.

Generalized Derivations of Hom-Lie Superalgebras

In this paper, we give some properties of the generalized derivation algebra $\mathrm{GDer}(T)$ of a Hom–Lie triple systems T. In particular, we prove that $\mathrm{GDer}(T) = \mathrm{QDer}(T) + \mathrm{QC}(T)$, the sum of the quasiderivation algebra and the quasicentroid. We also prove that $\mathrm{QDer}(T)$ can be embedded as derivations in a larger Hom–Lie triple system. General results on centroids of Hom–Lie triple systems are also developed in this paper.

Generalized Derivations of Hom–Lie Triple Systems

In this paper, we study (non-Abelian) extensions of a given hom-Lie color algebra and provide a geometrical interpretation of extensions. In particular, we characterize an extension of a hom-Lie co ...

Extensions of hom-lie color algebras

The main purpose of this paper is to define representations and a cohomology of Hom–Lie color algebras and to study some key constructions and properties. We describe Hartwig–Larsson–Silvestrov Theorem in the case of Γ-graded algebras, study one-parameter formal deformations, discuss α k -generalized derivations and provide examples.

Constructions and Cohomology of Hom–Lie Color Algebras

The main purpose of this paper is to define representations and a cohomology of color Hom-Lie algebras and to study some key constructions and properties. We describe Hartwig-Larsson-Silvestrov Theorem in the case of $\Gamma$-graded algebras, study one-parameter formal deformations, discuss $\alpha^{k}$-generalized derivation and provide examples.

Constructions and Cohomology of color Hom-Lie algebras

Hom-alternative, Hom-Malcev and Hom-Jordan superalgebras are $\mathbb{Z}_{2}$-graded generalizations of Hom-alternative, Hom-Malcev and Hom-Jordan algebras, which are Hom-type generalizations of alternative, Malcev and Jordan algebras. In this paper we prove that Hom-alternative superalgebras are Hom-Malcev-admissible and are also Hom-Jordan-admissible. Home-type generalizations of some well known identities in alternative superalgebras, including the $\mathbb{Z}_{2}$-graded Bruck-Kleinfled function are obtained.

K. Abdaoui

Papers

Constructions and Cohomology of Hom–Lie Color Algebras

Constructions and Cohomology of color Hom-Lie algebras

Hom-Alternative, Hom-Malcev and Hom-Jordan superalgebras