The purpose of this paper is to define the representation and the cohomology of Hom-Lie superalgebras. Moreover we study Central extensions and provide as application the computations of the derivations and second cohomology group of q-deformed Witt superalgebra.

The purpose of this paper is to compute the second adjoint cohomology group of q-deformed Witt superalgebras. They are Hom-Lie superalgebras obtained by q-deformation of Witt Lie superalgebra, that is one considers σ-derivations instead of classical derivations.

In this paper we define the left and right β-Richardson-Nijenhuis bracket. With this bracket we define a right, left and symmetric 2p-Hom-Leibniz algebras and their cohomology. Moreover with the β-RN bracket we study the extension of deformation of Hom-Leibniz algebra.

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In this paper, we generalize the results about generalized derivations of Lie algebras to the case of BiHom-Lie algebras. In particular we give the classification of generalized derivations of Heisenberg BiHom-Lie algebras. The definition of the generalized derivation depends on some parameters (λ, µ, γ) ∈ C 3 . In particular for (λ, µ, γ) = (1, 1, 1), we obtain classical concept of derivation of BiHom-Lie algebra and for (λ, µ, γ) = (1, 1, 0) we obtain the centroid of BiHom-Lie algebra. We give classifications of 2-dimensional BiHom-Lie algebra, centroides and derivations of 2-dimensional BiHom-Lie algebras.