Locally conformal SKT almost abelian Lie algebras

TL;DR

This paper characterizes and classifies six-dimensional almost abelian Lie algebras that admit locally conformal SKT Hermitian structures, extending the understanding of these geometric structures and their compatibility with other Hermitian types.

Contribution

It provides a complete classification of LCSKT structures on almost abelian Lie algebras in dimension six and explores their compatibility with other Hermitian structures.

Findings

Classified all 6D almost abelian Lie algebras with LCSKT structures.

Identified conditions for the existence of LCSKT structures on these algebras.

Analyzed compatibility between LCSKT and other Hermitian structures.

Abstract

A locally conformal SKT (shortly LCSKT) structure is a Hermitian structure $(J, g)$ whose Bismut torsion 3-form $H$ satisfies the condition $dH = \alpha \wedge H$, for some closed non-zero 1-form $\alpha$. This condition was introduced in [6] as a generalization of the SKT (or pluriclosed) condition $dH= 0$. In this paper, we characterize the almost abelian Lie algebras admitting a Hermitian structure $(J, g)$ such that $dH = \alpha \wedge H$, for some closed 1-form $\alpha$. As an application we classifiy LCSKT almost abelian Lie algebras in dimension $6$. Finally, we also study on almost abelian Lie algebras the compatibility between the LCSKT condition and other types of Hermitian structures.