Locally conformal SKT almost abelian Lie algebras

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.