Submaximally Symmetric Almost Quaternionic Structures

TL;DR

This paper determines the submaximal symmetry dimensions of almost quaternionic structures, showing that for non-flat cases, the symmetry dimension is $4n^2-4n+9$, achieved by specific structures.

Contribution

It computes the second-largest symmetry dimension for almost quaternionic structures and characterizes the structures realizing this symmetry.

Findings

Maximal symmetry for flat structures is $4n^2+8n+3$.

Submaximal symmetry dimension is $4n^2-4n+9$ for $n>1$.

Both torsion-free and Weyl-flat structures realize the submaximal symmetry.

Abstract

The symmetry dimension of a geometric structure is the dimension of its symmetry algebra. We investigate symmetries of almost quaternionic structures of quaternionic dimension $n$. The maximal possible symmetry is realized by the quaternionic projective space $\mathbb{H}P^n$, which is flat and has the symmetry algebra $\mathfrak{sl}(n+1,\mathbb{H})$ of dimension $4n^2+8n+3$. For non-flat almost quaternionic manifolds we compute the next biggest (submaximal) symmetry dimension. We show that it is equal to $4n^2-4n+9$ for $n>1$ (it is equal to 8 for $n=1$). This is realized both by a quaternionic structure (torsion--free) and by an almost quaternionic structure with vanishing quaternionic Weyl curvature.