Locally conformally Kähler structures on unimodular Lie groups

Abstract

We study left-invariant locally conformally Kähler 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.

17 pages

References in corpus (1)

• Compact Homogeneous Locally Conformally Kaehler Manifolds

Cited by in corpus (4)

• Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures
• Locally conformally Kähler structures on four-dimensional solvable Lie algebras
• Conformal Killing 2-forms on 4-dimensional manifolds
• Abelian Balanced Hermitian structures on unimodular Lie algebras