Quaternionic projective space on arvau

Wikipedia / Quaternionic projective space

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 quaternions Quaternionic projective space of dimension n is usually denoted by

and is a closed manifold of (real) dimension 4n. It is a homogeneous space for a Lie group 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 of a point can be written

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

In the language of group actions is the orbit space, the group of unit quaternions. then becomes a principal Sp(1)-bundle:

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

lies inside a complex Grassmannian.

Topology

Homotopy theory

's under inclusion, is the classifying space BS3. However, we do have that

It follows that rationally, i.e. after localisation of a space is an Eilenberg–Maclane space (cf. the example K(Z,2)). See rational homotopy theory.

Differential geometry

carries a natural Riemannian metric analogous to the Fubini-Study metric, 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

is the compact symplectic group.

Characteristic classes

, 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:

is its reduction mod 2.

Special cases

Quaternionic projective line

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. For the linear fractional transformations of an associative 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

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

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.