On $(\lambda ,\mu ,\gamma )$-derivations of BiHom-Lie algebras

Abstract

In this paper, we generalize the results about generalized derivations of Lie algebras to the case of BiHom-Lie algebras. In particular we give the classification of generalized derivations of Heisenberg BiHom-Lie algebras. The definition of the generalized derivation depends on some parameters $(\lambda ,\mu ,\gamma )\in {\mathbb{C}}^3.$ In particular for $(\lambda ,\mu ,\gamma )=(1,1,1)$, we obtain classical concept of derivation of BiHom-Lie algebra and for $(\lambda ,\mu ,\gamma )=(1,1,0)$ we obtain the centroid of BiHom-Lie algebra. We give classifications of $2$-dimensional BiHom-Lie algebra, centroides and derivations of $2$-dimensional BiHom-Lie algebras.