Quaternionic projective space: Difference between revisions

Revision as of 15:08, 23 September 2008

Latest revision as of 17:30, 5 June 2023

(33 intermediate revisions by 22 users not shown)

Line 1:

{{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

:<math>\mathbb{HP}^n</math>

:'''H'''''P''^''n''

and is a [[closed manifold]] of (real) dimension 4''4nn''. It is a [[homogeneous space]] for a [[Lie group]] action, in more than one way. The quaternionic projective line <math>\mathbb{HP}^1</math> is homeomorphic to the 4-sphere.

and is a [[closed manifold]] of (real) dimension ''4n''. It is a [[homogeneous space]] for a [[Lie group]] action, in more than one way.

==In coordinates==

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

:<math>[q_0,q_1,\ldots,q_n]</math>

:[''q''₀:''q''₁: ... :''q''_''n'']

where the ''q''<submath>''i''q_i</submath> 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

:<math>[cq_0,cq_1\ldots,cq_n]</math>.

:[''cq''₀:''cq''₁: ... :''cq''_''n''].

In the language of [[Group action (mathematics)|group action]]s, '''H'''''P''<supmath>''\mathbb{HP}^n''</supmath> is the [[orbit space]] of '''H'''<supmath>''\mathbb{H}^{n''+1}\setminus\{(0,\ldots,0)\}</supmath> by the action of '''<math>\mathbb{H'''*}^{\times}</math>, the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside '''H'''<supmath>''\mathbb{H}^{n''+1}</supmath> one may also regard '''H'''''P''<supmath>''\mathbb{HP}^{n''}</supmath> as the orbit space of ''S''<supmath>4''n''S^{4n+3}</supmath> by the action of <math>\text{Sp}(1)</math>, the group of unit quaternions.<ref>{{cite book |first=Gregory L. |last=Naber |chapter=Physical and Geometrical Motivation |title=Topology, Geometry and Gauge fields |publisher=Springer |series=Texts in Applied Mathematics |volume=25 |year=2011 |orig-year=1997 |isbn=978-1-4419-7254-5 |page=50 |doi=10.1007/978-1-4419-7254-5_0 |chapter-url=https://books.google.com/books?id=MObgBwAAQBAJ&pg=PR1}}</ref> The sphere ''S''<supmath>4''n''S^{4n+3}</supmath> then becomes a [[principal bundle|principal Sp(1)-bundle]] over '''H'''''P''<supmath>''\mathbb{HP}^n''</supmath>:

:<math>\mathrm{Sp}(1) \to S^{4n+3} \to \mathbb HP^n.</math>

:<math>\mathrm{Sp}(1) \to S^{4n+3} \to \mathbb {HP}^n.</math>

There is also a construction of '''H'''''P''^''n'' by means of two-dimensional complex subspaces of '''C'''^2''n'', meaning that '''H'''''P''^''n'' lies inside a complex [[Grassmannian]].

==Infinite-dimensional quaternionic projective space==

The space <math>\mathbb{HP}^{\infty}</math> is rationally K(Z,4) (cf. [[K(Z,2)]]). Please expand.

==Projective line==

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 [[Mobius group]] to the quaternion context with "linear fractional transformations". See [[inversive ring geometry]] for the uses of the projective line of the arbitrary [[ring (mathematics)|ring]].

This bundle is sometimes called a (generalized) [[Hopf fibration]].

==Quaternionic projective plane==

There is also a construction of '''H'''''P''<supmath>''\mathbb{HP}^{n''}</supmath> by means of two-dimensional complex subspaces of '''C'''<supmath>2''n''\mathbb{H}^{2n}</supmath>, meaning that '''H'''''P''<supmath>''\mathbb{HP}^{n''}</supmath> lies inside a complex [[Grassmannian]].

The 8-dimensional '''H'''''P''^''2'' 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]]

==Topology==

:'''H'''''P''^''n''/''U''(1)

===Homotopy theory===

The space <math>\mathbb{HP}^{\infty}</math>, defined as the union of all finite <math>\mathbb{HP}^n</math>'s under inclusion, is the [[classifying space]] ''BS''³. 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 \Q \cong \begin{cases} \Q & i = 4 \\ 0 & i \neq 4 \end{cases}</math>

It follows that rationally, i.e. after [[localisation of a space]], <math>\mathbb{HP}^\infty</math> is an [[Eilenberg–Maclane space]] <math>K(\Q,4)</math>. That is <math>\mathbb{HP}^{\infty}_{\Q} \simeq K(\Z, 4)_{\Q}.</math> (cf. the example [[K(Z,2)]]). See [[rational homotopy theory]].

In general, <math>\mathbb{HP}^n</math> has a cell structure with one cell in each dimension which is a multiple of 4, up to <math>4n</math>. Accordingly, its cohomology ring is <math>\Z[v]/v^{n+1}</math>, where <math>v</math> is a 4-dimensional generator. This is analogous to complex projective space. It also follows from rational homotopy theory that <math>\mathbb{HP}^n</math> has infinite homotopy groups only in dimensions 4 and <math>4n+3</math>.

==Differential geometry==

<math>\mathbb{HP}^n</math> carries a natural [[Riemannian metric]] analogous to the [[Fubini-Study metric]] on <math>\mathbb{CP}^n</math>, with respect to which it is a compact [[quaternion-Kähler symmetric space]] with positive curvature.

Quaternionic projective space can be represented as the coset space

:<math>\mathbb{HP}^n = \operatorname{Sp}(n+1)/\operatorname{Sp}(n)\times\operatorname{Sp}(1)</math>

where <math>\operatorname{Sp}(n)</math> is the compact [[symplectic group]].

===Characteristic classes===

Since <math>\mathbb{HP}^1=S^4</math>, its tangent bundle is stably trivial. The tangent bundles of the rest have nontrivial [[Stiefel–Whitney class|Stiefel–Whitney]] and [[Pontryagin class]]es. The total classes are given by the following formulas:

:<math>w(\mathbb{HP}^n) = (1+u)^{n+1}</math>

:<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>{{cite journal |first=R.H. |last=Szczarba |title=On tangent bundles of fibre spaces and quotient spaces |journal=American Journal of Mathematics |volume=86 |issue=4 |pages=685–697 |year=1964 |doi=10.2307/2373152 |jstor=2373152 |url=https://www.maths.ed.ac.uk/~v1ranick/papers/szczarba.pdf}}</ref>

==Special cases==

===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 [[MobiusMöbius group]] to the quaternion context with "[[linear fractional transformations". See [[inversive ring geometrytransformation]]s. forFor the useslinear offractional the projective linetransformations of thean arbitraryassociative [[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]].

Explicit expressions for coordinates for the 4-sphere can be found in the article on the [[Fubini–Study metric]].

===Quaternionic projective plane===

The 8-dimensional '''H'''''P''<supmath>''\mathbb{HP}^{2''}</supmath> 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]]

:<math>\mathbb{HP}^{2}/\mathrm{U}(1)</math>

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==

[[Category:Projective geometry]][[Category:Homogeneous spaces]][[Category:Quaternions]]

{{Reflist}}

==Further reading==

*{{cite journal |author-link=V. I. Arnol'd |first=V.I. |last=Arnol'd |title=Relatives of the Quotient of the Complex Projective Plane by the Complex Conjugation |journal=Tr. Mat. Inst. Steklova |volume=224 |pages=56–6 |year=1999 |citeseerx=10.1.1.50.6421 |url=http://mi.mathnet.ru/eng/tm/v224/p56 }} Treats the analogue of the result mentioned for quaternionic projective space and the 13-sphere.

* {{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&ndash;85

| year = 1947

| jstor = 20488472

}}

[[Category:Projective geometry]]

[[Category:Homogeneous spaces]]

[[Category:Quaternions]]