We study left-invariant generalized Kähler structures on almost abelian Lie groups, i.e., on solvable Lie groups with a codimension-one abelian normal subgroup. In particular, we classify six-dimensional almost abelian Lie groups which admit a left-invariant complex structure and establish which of those have a left-invariant Hermitian structure whose fundamental 2-form is \(\partial {{\overline{\partial }}}\)-closed. We obtain a classification of six-dimensional generalized Kähler almost abelian Lie groups and determine the six-dimensional compact almost abelian solvmanifolds admitting an invariant generalized Kähler structure. Moreover, we prove some results in relation to the existence of holomorphic Poisson structures and to the pluriclosed flow.

我们研究了几乎abelian Lie群上的左不变广义Kähler结构，即具有一维一阿贝尔正态子群的可解Lie群。尤其是，我们对允许左不变复数结构的六维几乎abelian Lie组进行分类，并确定其中哪些具有基本2形式为\（\ partial {{\ overline {\ partial} }} \）-关闭。我们获得了六维广义Kähler几乎阿贝尔Lie群的分类，并确定了包含不变广义Kähler结构的六维紧致几乎阿贝尔溶剂流形。此外，我们证明了与全纯泊松结构的存在和多闭流有关的一些结果。