  In the present paper, we describe two geometric notions, holomorphic Norden
structures and K\"{a}hler-Norden structures on Hom-Lie groups, and prove that
on Hom-Lie groups in the left invariant setting, these structures are related
to each other. We study K\"{a}hler-Norden structures with abelian complex
structures and give the curvature properties of holomorphic Norden structures
on Hom-Lie groups. Finally, we show that any left-invariant holomorphic Hom-Lie
group is a flat (holomorphic Norden Hom-Lie algebra carries a
Hom-Left-symmetric algebra) if its left-invariant complex structure (complex
structure) is abelian.
