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 <i>n</i>. The maximal possible symmetry is realized by the quaternionic projective space H<i>Pⁿ</i>, which is flat and has the symmetry algebra sl(<i>n</i> + 1, H) of dimension 4<i>n</i>² + 8<i>n</i> + 3. For non-flat almost quaternionic manifolds we compute the next biggest (submaximal) symmetry dimension. We show that it is equal to 4<i>n</i>²−4<i>n</i>+9 for <i>n</i> > 1 (it is equal to 8 for <i>n</i> = 1). This is realized both by a quaternionic structure (torsion–free) and by an almost quaternionic structure with vanishing quaternionic Weyl curvature.