SUBMAXIMALLY SYMMETRIC ALMOST QUATERNIONIC STRUCTURES

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 ℍP n , which is at and has the symmetry algebra $$ \mathfrak{sl}\left(n+1,\mathrm{\mathbb{H}}\right) $$ 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.