# Submaximally Symmetric Quaternion Hermitian Structures

## TL;DR

This paper determines the maximal and submaximal symmetry dimensions for almost quaternion-Hermitian structures, classifies all such structures with these symmetries, and explores their geometric properties.

## Contribution

It resolves the gap problem for symmetry dimensions in almost quaternion-Hermitian structures and classifies all structures with maximal and submaximal symmetries.

## Key findings

- Identified maximal and submaximal symmetry dimensions for these structures.
- Classified all structures with these symmetry dimensions.
- Studied geometric properties, including conformally quaternion-Kähler and quaternion-Kähler with torsion.

## Abstract

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-K\"ahler structures as well as quaternion-K\"ahler with torsion.