Kähler-Norden structures on Hom-Lie groups and Hom-Lie algebras

The aim of this paper is to describe two geometric notions, holomorphic Norden structures and Kähler-Norden structures on Hom-Lie groups, and study their relationships in the left invariant setting. We study Kähler-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.