Homogeneous Quaternionic Kahler Structures and Quaternionic Hyperbolic Space

Abstract

An explicit classification of homogeneous quaternionic Kähler structures by real tensors is derived and we relate this to the representationtheoretic description found by Fino. We then show how the quaternionic hyperbolic space HH(n) is characterised by admitting homogeneous structures of a particularly simple type. In the process we study the properties of different homogeneous models for HH(n).

Homogeneous pseudo-Riemannian structures of linear type

“…Hence, both Fino's classification [21,Lemma 5.1] and [14,Th. 3.15] extend to any signature of the metric.…”

“…Similarly to CH(n) (see Remark 3.13), the homogeneous structure S now defined by the vector field ξ on E + in (4.11) does not give the description of HH(n) as a solvable Lie group. A description of HH(n) as a solvable group can be given by a homogeneous quaternionic Kähler structure of type QK 134 (see [14,Prop. 5.3]).…”

“…. , QK 5 , and then they were also studied in other papers ( [6,12,13,14]). In the sequel we shall simply denote the class T i ⊕ T j by T i j , the class K i ⊕ K j by K i j , and so on.…”

“…4.3] (for the pseudo-Riemannian case see [26]), that of the complex hyperbolic space CH(n) in terms of homogeneous Kähler structures of type K 24 was obtained in [24,Th. 1.1], and that of the quaternionic hyperbolic space HH(n) by homogeneous quaternionic Kähler structures of type QK 123 (with nonzero projection to QK 3 ) was given in [14,Th. 1.1].…”

“…[14,15]), because the corresponding space is the space of sections of a vector bundle whose fibre dimension grows linearly with that of the base manifold in the three cases, that base manifold realising such a class in each of them.…”