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arXiv · 1607.02025

Submaximally Symmetric Almost Quaternionic Structures

Boris Kruglikov·Henrik Winther·Lenka Zalabova
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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.

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Boris Kruglikov, Henrik Winther, Lenka Zalabova. 2016-07-07. Submaximally Symmetric Almost Quaternionic Structures. https://arxiv.org/abs/1607.02025

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