Representations and cohomologies of relative Rota-Baxter Lie algebras and applications

In this paper, first we give the notion of a representation of a relative Rota-Baxter Lie algebra and introduce the cohomologies of a relative Rota-Baxter Lie algebra with coefficients in a representation. Then we classify abelian extensions of relative Rota-Baxter Lie algebras using the second cohomology group, and classify skeletal relative Rota-Baxter Lie 2-algebras using the third cohomology group as applications. At last, using the established general framework of representations and cohomologies of relative Rota-Baxter Lie algebras, we give the notion of representations of Rota-Baxter Lie algebras, which is consistent with representations of Rota-Baxter associative algebras in the literature, and introduce the cohomologies of Rota-Baxter Lie algebras with coefficients in a representation. Applications are also given to classify abelian extensions of Rota-Baxter Lie algebras and skeletal Rota-Baxter Lie 2-algebras.