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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 '''H'''. Quaternionic projective space of dimension ''n'' is usually denoted by
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| :<math>\mathbb{HP}^n</math>
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| and is a [[closed manifold]] of (real) dimension 4''n''. It is a [[homogeneous space]] for a [[Lie group]] action, in more than one way.
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| ==In coordinates==
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| Its direct construction is as a special case of the [[projective space over a division algebra]]. The [[homogeneous coordinates]] of a point can be written
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| :<math>[q_0,q_1,\ldots,q_n]</math>
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| where the <math>q_i</math> are quaternions, not all zero. Two sets of coordinates represent the same point if they are 'proportional' by a left multiplication by a non-zero quaternion ''c''; that is, we identify all the
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| :<math>[cq_0,cq_1\ldots,cq_n]</math>.
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| In the language of [[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>\mathbb{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>\mathbb{S}^{4n+3}</math> then becomes a [[principal bundle|principal Sp(1)-bundle]] over <math>\mathbb{HP}^n</math>:
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| :<math>\mathrm{Sp}(1) \to \mathbb{S}^{4n+3} \to \mathbb{HP}^n.</math>
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| 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]].
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| ==Projective line==
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| The one-dimensional projective space over '''H''' 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 transformations".
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| 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'').
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| 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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| ==Infinite-dimensional quaternionic projective space==
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| The space <math>\mathbb{HP}^{\infty}</math> is the [[classifying space]] BS<sup>3</sup>. The homotopy groups of <math>\mathbb{HP}^{\infty}</math> are given by <math>\pi_i(\mathbb{HP}^{\infty}) = \pi_i(BS^3) \cong \pi_{i-1}(S^3)</math>. These groups are known to be very complex and in particular they are non-zero for infinitely many values of <math>i</math>. However, we do have that <math>\pi_i(\mathbb{HP}^\infty) \otimes \mathbb{Q} \cong \mathbb{Q}</math> if <math>i = 4 </math> and <math>\pi_i(\mathbb{HP}^\infty) \otimes \mathbb{Q} = 0 </math> if <math>i \neq 4 </math>. It follows that rationally, i.e. after [[localisation of a space]], <math>\mathbb{HP}^\infty</math> is an [[Eilenberg–Maclane space]] <math>K(\mathbb{Q},4)</math>. That is <math>\mathbb{HP}^{\infty}_{\mathbb{Q}} \simeq K(\mathbb{Z}, 4)_{\mathbb{Q}}</math>. (cf. the example [[K(Z,2)]]). See [[rational homotopy theory]].
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| ==Quaternionic projective plane==
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| The 8-dimensional <math>\mathbb{HP}^{2}</math> has a [[circle action]], by the group of complex scalars of absolute value 1 acting on the other side (so on the right, as the convention for the action of ''c'' above is on the left). Therefore the [[quotient manifold]]
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| <math>\mathbb{HP}^{2}/\mathrm{U}(1)</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]].
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| *[[V. I. Arnol'd]], ''Relatives of the Quotient of the Complex Projective Plane by the Complex Conjugation'', Tr. Mat. Inst. Steklova, 1999, Volume 224, Pages 56–67. Treats the analogue of the result mentioned for quaternionic projective space and the 13-sphere.
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| [[Category:Projective geometry]]
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| [[Category:Homogeneous spaces]]
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| [[Category:Quaternions]]
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The name of the author is Numbers. California is our birth place. I am a meter reader. One of the issues he loves most is ice skating but he is having difficulties to find time for it.
Look into my page ... www.pinaydiaries.com