Cohomologies of modified Rota-Baxter Lie algebras with derivations and applications

Abstract: In this paper, first, we introduce a notion of modified Rota-Baxter Lie algebras of weight $\mathrm{\lambda}$ with derivations (or simply modified Rota-Baxter LieDer pairs) and their representations. Moreover, we investigate cohomologies of a modified Rota-Baxter LieDer pairs with coefficients in a suitable representation. As applications, we study formal one-parameter deformations and abelian extensions of modified Rota-Baxter LieDer pairs.