  In this paper, we provide a unified approach to study the cohomology theories
and deformation theories of various types of operators in the category of Lie
algebras, including modified $r$-matrices, crossed homomorphisms, derivations,
homomorphisms, relative Rota-Baxter operators, twisted Rota-Baxter operators,
Reynolds operators and deformation maps of matched pairs of Lie algebras. The
main ingredients are quasi-twilled Lie algebras. We introduce two types of
deformation maps of a quasi-twilled Lie algebra. Deformation maps of type I
unify modified $r$-matrices, crossed homomorphisms, derivations and
homomorphisms between Lie algebras, while deformation maps of type II unify
relative Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds
operators and deformation maps of matched pairs of Lie algebras. We further
give the controlling algebras and cohomologies of these two types of
deformation maps, which not only recover the existing results for crossed
homomorphisms, derivations, homomorphisms, relative Rota-Baxter operators,
twisted Rota-Baxter operators and Reynolds operators, but also leads to some
new results which are unable to solve before, e.g. the controlling algebras and
cohomologies of modified $r$-matrices and deformation maps of matched pairs of
Lie algebras.
