Maurer-Cartan characterization and cohomology of compatible LieDer and AssDer pairs

A LieDer pair (respectively, an AssDer pair) is a Lie algebra equipped with a derivation (respectively, an associative algebra equipped with a derivation). A couple of LieDer pair structures on a vector space are called Compatible LieDer pairs (respectively, compatible AssDer pairs) if any linear combination of the underlying structure maps is still a LieDer pair (respectively, AssDer pair) structure. In this paper, we study compatible AssDer pairs, compatible LieDer pairs, and their cohomologies. We also discuss other compatible structures such as compatible dendriform algebras with derivations, compatible Zinbiel algebras with derivations, and compatible pre-LieDer pairs. We describe a relationship among these compatible structures using specific tools like Rota-Baxter operators, endomorphism operators, and the commutator bracket.