  In this paper, first we show that there is a Hom-Lie algebra structure on the
set of $(\sigma,\sigma)$-derivations of a commutative algebra. Then we
construct dual representations of a representation of a Hom-Lie algebra. We
introduce the notions of a Manin triple for Hom-Lie algebras and a purely
Hom-Lie bialgebra. Using the coadjoint representation, we show that there is a
one-to-one correspondence between Manin triples for Hom-Lie algebras and purely
Hom-Lie bialgebras. Finally, we study coboundary purely Hom-Lie bialgebras and
construct solutions of the classical Hom-Yang-Baxter equations in some special
Hom-Lie algebras using Hom-$\mathcal O$-operators.
