We consider and resolve the gap problem for almost quaternion-Hermitian structures, i.e. we determine the maximal and submaximal symmetry dimensions, both for Lie algebras and Lie groups, in the class of almost quaternion-Hermitian manifolds. We classify all structures with such symmetry dimensions. Geometric properties of the submaximally symmetric spaces are studied, in particular, we identify locally conformally quaternion-Kähler structures as well as quaternion-Kähler with torsion.

中文翻译：

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次极大对称四元数厄米特结构

我们考虑并解决了几乎四元数-厄米特结构的间隙问题，即我们确定了几乎四元数-厄米特流形类中李代数和李群的最大和次最大对称维数。我们对所有具有这种对称尺寸的结构进行分类。研究了次最大对称空间的几何性质，特别是，我们识别了局部共形四元数-Kähler 结构以及具有扭转的四元数-Kähler。