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In this paper, we introduce the notion of a (regular) Hom-Lie group. Weassociate a Hom-Lie algebra to a Hom-Lie group and show that every regularHom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) mapfrom the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discussthe universality of this Hexp map. We also describe a Hom-Lie group action on asmooth manifold. Subsequently, we give the notion of an adjoint representationof a Hom-Lie group on its Hom-Lie algebra. At last, we integrate the Hom-Liealgebra $(\mathfrak{gl}(V),[\cdot,\cdot],\mathsf{Ad})$, and the derivationHom-Lie algebra of a Hom-Lie algebra.

Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation