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The additive (generalized) $\xi$-Lie derivations on prime algebras arecharacterized. It is shown, under some suitable assumption, that an additivemap $L$ is an additive (generalized) Lie derivation if and only if it is thesum of an additive (generalized) derivation and an additive map from thealgebra into its center vanishing all commutators; is an additive (generalized)$\xi$-Lie derivation with $\xi\not=1$ if and only if it is an additive(generalized) derivation satisfying $L(\xi A)=\xi L(A)$ for all $A$. Theseresults are then used to characterize additive (generalized) $\xi$-Liederivations on several operator algebras such as Banach space standard operatoralgebras and von Neumman algebras.

Additive Lie ($\xi$-Lie) Derivations and Generalized Lie ($\xi$-Lie) Derivations on Prime Algebras