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

Let ℜ be a unital algebra over a field (Formula presented.) with (Formula presented.), and let (Formula presented.) be linear mappings. We say that (Formula presented.) is a (Formula presented.) -derivation if (Formula presented.) The mapping (Formula presented.) is said to be a Lie (Formula presented.) -derivation if (Formula presented.) where (Formula presented.) denotes the Lie product. In this paper, we prove that if every Lie (Formula presented.) -derivation on ℜ is necessarily a (Formula presented.) -derivation, then the same property holds for the tensor product algebra (Formula presented.), where ℑ is any commutative unital algebra. Moreover, every Lie (Formula presented.) -derivation of a semiprime algebra is a (Formula presented.) -derivation. As a consequence, Lie derivations on tensor products of semiprime algebras with commutative algebras reduce to derivations in the classical sense.