  A difference Lie group is a Lie group equipped with a difference operator,
equivalently a crossed homomorphism with respect to the adjoint action. In this
paper, first we introduce the notion of a representation of a difference Lie
group, and establish the relation between representations of difference Lie
groups and representations of difference Lie algebras via differentiation and
integration. Then we introduce a cohomology theory for difference Lie groups
and justify it via the van Est theorem. Finally, we classify abelian extensions
of difference Lie groups using the second cohomology group as applications.
