Cohomologies and deformations of weighted Rota-Baxter Lie algebras and associative algebras with derivations

The purpose of the present paper is to investigate cohomologies and deformations of weighted Rota-Baxter Lie algebras as well as weighted Rota-Baxter associative algebras with derivations. First we introduce a notion of weighted Rota-Baxter LieDer and weighted Rota-Baxter AssDer pairs. Then we construct cohomologies of weighted Rota-Baxter LieDer pairs, weighted Rota-Baxter AssDer pairs and we discuss the relation between their cohmologies. Finally, as an application, we study deformations of both of them.