Hom-Lie Algebras with Derivations

In this paper, we first introduce the notion of an HLieDer triple, which includes a Hom-Lie algebra and a derivation. We define a cohomology theory for HLieDer triples with coefficients in a representation. We study central extensions of an HLieDer triple. Finally, we consider homotopy derivations on HLieb infinity algebras and 2-derivations on Hom-Lie 2-algebras, and we prove that the category of 2-term HLieb infinity algebras with homotopy derivations and the category of Hom-Lie 2-algebras with 2-derivations are equivalent.