Biderivations of Hom-Lie algebras and superalgebras

Abstract

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 α-derivations and α-biderivations for Hom-Lie algebras and superalgebras, and we establish a close relation between α-derivations and α-biderivations. As an illustration, we prove that the q-deformed W (2, 2)-algebra, the q-deformed Witt algebra and superalgebra have no nontrivial α-biderivations. Finally, we present an example of Hom-Lie superalgebras with nontrivial α-super-derivations and biderivations.