Study on Lie {ξ,ζ}-Derivations on Tensor Products of Algebras

Let ℜ be a unital algebra over a field k with char(k)≠2, and let ϝ,ξ,ζ:ℜ→ℜ be linear mappings. We say that ϝ is a {ξ,ζ}-derivation if ϝ(ϑς)=ξ(ϑ)ς+ϑζ(ς)=ζ(ϑ)ς+ϑξ(ς)forallϑ,ς∈ℜ. The mapping ϝ is said to be a Lie {ξ,ζ}-derivation if ϝ([ϑ,ς])=[ξ(ϑ),ς]+[ϑ,ζ(ς)]forallϑ,ς∈ℜ, where [ϑ,ς]=ϑς−ςϑ denotes the Lie product. In this paper, we prove that if every Lie {ξ,ζ}-derivation on ℜ is necessarily a {ξ,ζ}-derivation, then the same property holds for the tensor product algebra ℜ⊗ℑ, where ℑ is any commutative unital algebra. Moreover, every Lie {ξ,ζ}-derivation of a semiprime algebra is a {ξ,ζ}-derivation. As a consequence, Lie derivations on tensor products of semiprime algebras with commutative algebras reduce to derivations in the classical sense.