Hermitian geometry of Lie algebras with abelian ideals of codimension $2$

TL;DR

This paper classifies Hermitian metrics on unimodular Lie algebras with a specific abelian ideal, focusing on Bismut Kähler-like and torsion-parallel metrics, advancing understanding of their geometric structures.

Contribution

It provides a complete classification of Bismut Kähler-like and torsion-parallel metrics on Lie algebras with a J-invariant abelian ideal of codimension two.

Findings

Classification of Bismut Kähler-like metrics

Classification of Bismut torsion-parallel metrics

New insights into Hermitian geometry of Lie algebras

Abstract

We examine Hermitian metrics on unimodular Lie algebras which contains a $J$-invariant abelian ideal of codimension two, and give a classification for all Bismut K\"ahler-like and all Bismut torsion-parallel metrics on such Lie algebras.