ON (α, β, γ)-DERIVATIONS OF LIE SUPERALGEBRAS

Abstract

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.