Complex and Kahler structures on hom-Lie algebras

TL;DR

This paper introduces complex, Hermitian, and Kahler structures on hom-Lie algebras, explores their properties, and constructs phase spaces using these structures, including a proof of non-existence in dimension two.

Contribution

It defines new geometric structures on hom-Lie algebras and constructs phase spaces, expanding the understanding of their geometric and algebraic properties.

Findings

No proper complex or Hermitian hom-Lie algebra exists in dimension two.

Phase spaces can be constructed from hom-left symmetric and Kahler hom-Lie algebras.

Complex and Kahler structures on hom-Lie algebras are feasible and can be explicitly realized.

Abstract

Complex and Hermitian structures on hom-Lie algebras are introduced and some examples of these structures are presented. Also, it is shown that there not exists a proper complex (Hermitian) home-Lie algebra of dimension two. Then using a hom-left symmetric algebra, a phase space is provided and then a complex structure on it, is presented. Finally, the notion of Kahler hom-Lie algebra is introduced and then using a Kahler hom-Lie algebra, a phase space is constructed.