This paper is primarily concerned with (α, β, γ)-derivations of finite-dimensional Lie superalgebras over the field of complex numbers. Some properties of (α, β, γ)-derivations of the Lie superalgebras are obtained. In particular, two examples for (α, β, γ)-derivations of low-dimensional non-simple Lie superalgebras are presented and the super-spaces of (α, β, γ)-derivations for simple Lie superalgebras are determined. Using certain complex parameters we generalize the concept of cohomology cocycles of Lie superalgebras. A special case for the generalization of 1-cocycles with respect to the adjoint representation is exactly (α, β, γ)-derivations. Furthermore, two-dimensional twisted cocycles of the adjoint representation are investigated in detail.

No abstract

This paper is concerned with the natural filtration of Lie superalgebraS(n,m)of special type over a field of prime characteristic. We first construct the modular Lie superalgebraS(n,m). Then we prove that the natural filtration ofS(n,m)is invariant under its automorphisms.

We study the natural filtrations of the finite-dimensional modular Lie superalgebras W (n, m) and H(n, m). In particular, the natural filtrations which are invariant relative to the automorphisms of the Lie superalgebras are employed in order to characterize the Lie superalgebras themselves.