Locally conformally balanced metrics on almost abelian Lie algebras

TL;DR

This paper characterizes locally conformally balanced metrics on almost abelian Lie algebras, classifies six-dimensional cases, and explores compatibility with other special Hermitian structures, including hyperk"ahler metrics.

Contribution

It provides a comprehensive classification of locally conformally balanced and hyperk"ahler structures on almost abelian Lie algebras, especially in six dimensions.

Findings

Classification of six-dimensional almost abelian Lie algebras with locally conformally balanced metrics.

Characterization of compatibility between different special Hermitian metrics.

Identification of almost abelian Lie algebras admitting locally conformally hyperk"ahler structures.

Abstract

We study locally conformally balanced metrics on almost abelian Lie algebras, namely solvable Lie algebras admitting an abelian ideal of codimension one, providing characterizations in every dimension. Moreover, we classify six-dimensional almost abelian Lie algebras admitting locally conformally balanced metrics and study some compatibility results between different types of special Hermitian metrics on almost abelian Lie groups and their compact quotients. We end by classifying almost abelian Lie algebras admitting locally conformally hyperk\"ahler structures.