  The notion of a matched pair of Lie algebras was introduced in the study of
Lie bialgebras and Poisson-Lie groups. In this paper, we introduce
representations and cohomology of a matched pair of Lie algebras. We show that
there is a morphism from the cohomology of a Lie bialgebra to the cohomology of
the corresponding matched pair of Lie algebras. Our cohomology is also useful
to study infinitesimal deformations and abelian extensions. In the last part,
we define the notion of a matched pair of $L_\infty$-algebras and construct the
corresponding bicrossed product. Finally, we show that a matched pair of
skeletal $L_\infty$-algebras is closely related to the cohomology introduced in
the paper.
