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A LieDer pair (respectively, an AssDer pair) is a Lie algebra equipped with aderivation (respectively, an associative algebra equipped with a derivation). Acouple of LieDer pair structures on a vector space are called Compatible LieDerpairs (respectively, compatible AssDer pairs) if any linear combination of theunderlying structure maps is still a LieDer pair (respectively, AssDer pair)structure. In this paper, we study compatible AssDer pairs, compatible LieDerpairs, and their cohomologies. We also discuss about other compatiblestructures such as compatible dendriform algebras with derivations, compatiblezinbiel algebras with derivations, and compatible pre-LieDer pairs. We describea relationship amongst these compatible structures using specific tools likeRota-Baxter operators, endomorphism operators, and the commutator bracket.

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