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

Cohomologies of difference Lie groups and van Est theorem