K\"{a}hler-Norden structures on Hom-Lie group and Hom-Lie algebras

TL;DR

This paper explores geometric structures called holomorphic Norden and K"{a}hler-Norden on Hom-Lie groups, analyzing their properties, relationships, and curvature, especially in the context of abelian complex structures and flatness conditions.

Contribution

It introduces and studies K"{a}hler-Norden structures on Hom-Lie groups, establishing their relation to holomorphic Norden structures and characterizing flatness in the abelian case.

Findings

K"{a}hler-Norden and holomorphic Norden structures are related on Hom-Lie groups.

Holomorphic Norden structures are flat if the complex structure is abelian.

Curvature properties are characterized for these structures on Hom-Lie groups.

Abstract

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.