In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s
Quaternionic projective space of dimension ''n'' is usually denoted by
:
and is a
closed manifold
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only connected one-dimensional example ...
of (real) dimension 4''n''. It is a
homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
for a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
action, in more than one way. The quaternionic projective line
is homeomorphic to the 4-sphere.
In coordinates
Its direct construction is as a special case of the
projective space over a division algebra. The
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
of a point can be written
:
orbit space
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
of
\mathbb^\setminus\ by the action of
\mathbb^, the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside
\mathbb^ one may also regard
\mathbb^ as the orbit space of
S^ by the action of
\text(1), the group of unit quaternions. The sphere
S^ then becomes a
principal Sp(1)-bundle over
\mathbb^n:
:
\mathrm(1) \to S^ \to \mathbb^n.
This bundle is sometimes called a (generalized)
Hopf fibration.
There is also a construction of
\mathbb^ by means of two-dimensional complex subspaces of
\mathbb^, meaning that
\mathbb^ lies inside a complex
Grassmannian.
Topology
Homotopy theory
The space
\mathbb^, defined as the union of all finite
\mathbb^n's under inclusion, is the
classifying space ''BS''
3. The homotopy groups of
\mathbb^ are given by
\pi_i(\mathbb^) = \pi_i(BS^3) \cong \pi_(S^3). These groups are known to be very complex and in particular they are non-zero for infinitely many values of
i. However, we do have that
:
\pi_i(\mathbb^\infty) \otimes \Q \cong \begin \Q & i = 4 \\ 0 & i \neq 4 \end
It follows that rationally, i.e. after
localisation of a space,
\mathbb^\infty is an
Eilenberg–Maclane space K(\Q,4). That is
\mathbb^_ \simeq K(\Z, 4)_. (cf. the example
K(Z,2)). See
rational homotopy theory
In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by and . This simplification of homo ...
.
In general,
\mathbb^n has a cell structure with one cell in each dimension which is a multiple of 4, up to
4n. Accordingly, its cohomology ring is
\Z v^, where
v is a 4-dimensional generator. This is analogous to complex projective space. It also follows from rational homotopy theory that
\mathbb^n has infinite homotopy groups only in dimensions 4 and
4n+3.
Differential geometry
\mathbb^n carries a natural
Riemannian metric analogous to the
Fubini-Study metric on
\mathbb^n, with respect to which it is a compact
quaternion-Kähler symmetric space In differential geometry, a quaternion-Kähler symmetric space or Wolf space is a quaternion-Kähler manifold which, as a Riemannian manifold, is a Riemannian symmetric space. Any quaternion-Kähler symmetric space with positive Ricci curvature is ...
with positive curvature.
Quaternionic projective space can be represented as the coset space
:
\mathbb^n = \operatorname(n+1)/\operatorname(n)\times\operatorname(1)
where
\operatorname(n) is the compact
symplectic group.
Characteristic classes
Since
\mathbb^1=S^4, its tangent bundle is stably trivial. The tangent bundles of the rest have nontrivial
Stiefel–Whitney and
Pontryagin classes. The total classes are given by the following formulas:
:
w(\mathbb^n) = (1+u)^
:
p(\mathbb^n) = (1+v)^ (1+4v)^
where
v is the generator of
H^4(\mathbb^n;\Z) and
u is its reduction mod 2.
Special cases
Quaternionic projective line
The one-dimensional projective space over
\mathbb 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
Moebius, Möbius or Mobius may refer to:
People
* August Ferdinand Möbius (1790–1868), German mathematician and astronomer
* Theodor Möbius (1821–1890), German philologist
* Karl Möbius (1825–1908), German zoologist and ecologist
* Paul ...
to the quaternion context with
linear fractional transformations. For the linear fractional transformations of an associative
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
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
In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, on a complex projective space CP''n'' endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Edua ...
.
Quaternionic projective plane
The 8-dimensional
\mathbb^ 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
:
\mathbb^/\mathrm(1)
may be taken, writing
U(1) for the
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \ ...
. 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
Further reading
* 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
The ''Proceedings of the Royal Irish Academy'' (''PRIA'') is the journal of the Royal Irish Academy, founded in 1785 to promote the study of science, polite literature
Polite may refer to:
* Politeness
* ''Polite'' (magazine), an American humo ...
, volume = 51
, pages = 67–85
, year = 1947
, jstor = 20488472
Projective geometry
Homogeneous spaces
Quaternions