Additive Lie (ξ-Lie) derivations and generalized Lie (ξ-Lie) derivations on prime algebras

The additive (generalized) ξ-Lie derivations on prime algebras are characterized. It is shown, under some suitable assumptions, that an additive map L is an additive generalized Lie derivation if and only if it is the sum of an additive generalized derivation and an additive map from the algebra into its center vanishing all commutators; is an additive (generalized) ξ-Lie derivation with ξ ≠1 if and only if it is an additive (generalized) derivation satisfying L(ξA) = ξL(A) for all A. These results are then used to characterize additive (generalized) ξ-Lie derivations on several operator algebras such as Banach space standard operator algebras and von Neumman algebras.