  Using crossed homomorphisms, we show that the category of weak
representations (resp. admissible representations) of Lie-Rinehart algebras
(resp. Leibniz pairs) is a left module category over the monoidal category of
representations of Lie algebras. In particular, the corresponding bifunctor of
monoidal categories is established to give new weak representations (resp.
admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs).
This generalizes and unifies various existing constructions of representations
of many Lie algebras by using this new bifunctor. We construct some crossed
homomorphisms in different situations and use our actions of monoidal
categories to recover some known constructions of representations of various
Lie algebras, also to obtain new representations for generalized Witt algebras
and their Lie subalgebras. The cohomology theory of crossed homomorphisms
between Lie algebras is introduced and used to study linear deformations of
crossed homomorphisms.
