  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.
