Locally conformally K\"ahler structures on unimodular Lie groups

TL;DR

This paper investigates the properties of locally conformally K"ahler structures on Lie groups, focusing on unimodular cases and special complex structures, revealing restrictions and classifications for these geometric structures.

Contribution

It provides new results on the existence and classification of unimodular Lie algebras with locally conformally K"ahler structures, especially under bi-invariant or abelian complex structures.

Findings

No unimodular Lie algebra admits bi-invariant complex structures with these properties.

Unimodular Lie algebras with abelian complex structures are isomorphic to a product of nd a Heisenberg algebra.

Abstract

We study left-invariant locally conformally K\"ahler structures on Lie groups, or equivalently, on Lie algebras. We give some properties of these structures in general, and then we consider the special cases when its complex structure is bi-invariant or abelian. In the former case, we show that no such Lie algebra is unimodular, while in the latter, we prove that if the Lie algebra is unimodular, then it is isomorphic to the product of $\mathbb R$ and a Heisenberg Lie algebra.