MSC:17A70 17A30 16A03 17B68 Keywords:Hom-Lie superalgebra Hom-associative superalgebra Hom-Lie admissible superalgebra Lie admissible superalgebra q-Witt superalgebra The purpose of this paper is to study Hom-Lie superalgebras, that is a superspace with a bracket for which the superJacobi identity is twisted by a homomorphism. This class is a particular case of Γ -graded quasi-Lie algebras introduced by Larsson and Silvestrov. In this paper, we characterize Hom-Lie admissible superalgebras and provide a construction theorem from which we derive a one parameter family of Hom-Lie superalgebras deforming the orthosymplectic Lie superalgebra. Also, we prove a Z 2 -graded version of a Hartwig-Larsson-Silvestrov Theorem which leads us to a construction of a q-deformed Witt superalgebra.

The aim of this paper is to provide cohomologies of n-ary Hom-Nambu-Lie algebras governing central extensions and one parameter formal deformations. We generalize to n-ary algebras the notions of derivations and representation introduced by Sheng for Hom-Lie algebras. Also we show that a cohomology of n-ary Hom-Nambu-Lie algebras could be derived from the cohomology of Hom-Leibniz algebras.

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 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 -graded algebras, study one-parameter formal deformations, discuss k -generalized derivations and provide examples.

In this paper we construct ternary q-Virasoro-Witt algebras which q-deform the ternary Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos using su(1, 1) enveloping algebra techniques. The ternary Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos depend on a parameter and are not Nambu-Lie algebras for all but finitely many values of this parameter. For the parameter values for which the ternary Virasoro-Witt algebras are Nambu-Lie, the corresponding ternary q-Virasoro-Witt algebras constructed in this article are also Hom-Nambu-Lie because they are obtained from the ternary Nambu-Lie algebras using the composition method. For other parameter values this composition method does not yield Hom-Nambu Lie algebra structure for q-Virasoro-Witt algebras. We show however, using a different construction, that the ternary Virasoro-Witt algebras of Curtright, Fairlie and Zachos, as well as the general ternary q-Virasoro-Witt algebras we construct, carry a structure of ternary Hom-Nambu-Lie algebra for all values of the involved parameters.