Let (H,?) be a monoidal Hom-Hopf algebra and HH HYD the Hom-Yetter-Drinfeld
category over (H,?). Then in this paper, we first introduce the definition
of braided Hom-Lie algebras and show that each monoidal Hom-algebra in HH
HYD gives rise to a braided Hom-Lie algebra. Second, we prove that if (A,?)
is a sum of two H-commutative monoidal Hom-subalgebras, then the commutator
Hom-ideal [A,A] of A is nilpotent. Also, we study the central invariant of
braided Hom-Lie algebras as a generalization of generalized Lie algebras.
Finally, we obtain a construction of the enveloping algebras of braided
Hom-Lie algebras and show that the enveloping algebras are H-cocommutative
Hom-Hopf algebras.