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In [[mathematics]], '''quaternionic projective space''' is an extension of the ideas of [[real projective space]] and [[complex projective space]], to the case where coordinates lie in the ring of [[quaternion]]s <math>\mathbb{H}.</math> Quaternionic projective space of dimension ''n'' is usually denoted by
:<math>\mathbb{HP}^n</math>
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:<math>[cq_0,cq_1\ldots,cq_n]</math>.
In the language of [[Group action (mathematics)|group action]]s, <math>\mathbb{HP}^n</math> is the [[orbit space]] of <math>\mathbb{H}^{n+1}\setminus\{(0,\ldots,0)\}</math> by the action of <math>\mathbb{H}^{\times}</math>, the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside <math>\mathbb{H}^{n+1}</math> one may also regard <math>\mathbb{HP}^{n}</math> as the orbit space of <math>S^{4n+3}</math> by the action of <math>\text{Sp}(1)</math>, the group of unit quaternions.<ref>Gregory L. Naber, ''Topology, geometry, and gauge fields: foundations'' (1997), p. 50.</ref> The sphere <math>S^{4n+3}</math> then becomes a [[principal bundle|principal Sp(1)-bundle]] over <math>\mathbb{HP}^n</math>:
:<math>\mathrm{Sp}(1) \to S^{4n+3} \to \mathbb{HP}^n.</math>
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This bundle is sometimes called a (generalized) [[Hopf fibration]].
There is also a construction of <math>\mathbb{HP}^{n}</math> by means of two-dimensional complex subspaces of <math>\mathbb{H}^{2n}</math>, meaning that <math>\mathbb{HP}^{n}</math> lies inside a complex [[Grassmannian]].
==Topology==
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===Quaternionic projective line===
The one-dimensional projective space over <math>\mathbb{H}</math> is called the "projective line" in generalization of the [[complex projective line]]. For example, it was used (implicitly) in 1947 by P. G. Gormley to extend the [[Möbius group]] to the quaternion context with [[linear fractional transformation]]s. For the linear fractional transformations of an associative [[ring (mathematics)|ring]] with 1, see [[projective line over a ring]] and the homography group GL(2,''A'').
From the topological point of view the quaternionic projective line is the 4-sphere, and in fact these are [[diffeomorphic]] manifolds. The fibration mentioned previously is from the 7-sphere, and is an example of a [[Hopf fibration]].
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