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{{Short description|Concept in mathematics}}
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
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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>{{cite book |first=Gregory L. |last=Naber
:<math>\mathrm{Sp}(1) \to S^{4n+3} \to \mathbb{HP}^n.</math>
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:<math>p(\mathbb{HP}^n) = (1+v)^{2n+2} (1+4v)^{-1} </math>
where <math>v</math> is the generator of <math>H^4(\mathbb{HP}^n;\Z)</math> and <math>u</math> is its reduction mod 2.<ref>
==Special cases==
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==Further reading==
*
* {{Citation
| last = Gormley
| first = P.G.
| title = Stereographic projection and the linear fractional group of transformations of quaternions
| journal = [[Proceedings of the Royal Irish Academy, Section A]]
| volume = 51
| pages = 67–85
| year = 1947
| jstor = 20488472
}}
[[Category:Projective geometry]]
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