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:[''cq''<sub>0</sub>:''cq''<sub>1</sub>: ... :''cq''<sub>''n''</sub>].
In the language of [[group action]]s, '''H'''''P''<sup>''n''</sup> is the [[orbit space]] of '''H'''<sup>''n''+1</sup>-(0, ..., 0) by the action of '''H'''*, the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside '''H'''<sup>''n''+1</sup> one may also regard '''H'''''P''<sup>''n''</sup> as the orbit space of ''S''<sup>4''n''+3</sup> by the action of Sp(1), the group of unit quaternions.<ref>Gregory L. Naber, ''Topology, geometry, and gauge fields: foundations'' (1997), p. 50.</ref> The sphere ''S''<sup>4''n''+3</sup> then becomes a [[principal bundle|principal Sp(1)-bundle]] over '''H'''''P''<sup>''n''</sup>:
:<math>\mathrm{Sp}(1) \to S^{4n+3} \to \mathbb HP^n.</math>
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may be taken, writing [[U(1)]] for the [[circle group]]. It has been shown that this quotient is the 7-[[sphere]], a result of [[Vladimir Arnold]] from 1996, later rediscovered by [[Edward Witten]] and [[Michael Atiyah]].
==References==
{{Reflist}}
[[Category:Projective geometry]]
[[Category:Homogeneous spaces]]
[[Category:Quaternions]]
[[nl:Quaternionische projectieve ruimte]]
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