Biderivations of Hom-Lie algebras and superalgebras

In this paper, we introduce the notions of biderivations and linear commuting maps of Hom-Lie algebras and superalgebras. Then we compute biderivations of the q-deformed W(2,2) algebra, q-deformed Witt algebra and superalgebras by elementary and direct calculations. As an application, linear commuting maps on these algebras are characterized. Also, we introduce the notions of {alpha}-derivations and {alpha}-biderivations for Hom-Lie algebras and superal- gebras, and we establish a close relation between {alpha}-derivations and {alpha}-biderivations. As an illustration, we prove that the q-deformed W(2;2)-algebra, the q-deformed Witt algebra and superalgebra have no nontrivial {alpha}-biderivations. Finally, we present an example of Hom-Lie superalgebras with nontrivial {alpha}-super-derivations and biderivations.