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 action, in more than one way.
The quaternionic projective line
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 where the
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,
, the multiplicative group of non-zero quaternions.
By first projecting onto the unit sphere inside
: This bundle is sometimes called a (generalized) Hopf fibration.
by means of two-dimensional complex subspaces of
lies inside a complex Grassmannian.
These groups are known to be very complex and in particular they are non-zero for infinitely many values of
However, we do have that It follows that rationally, i.e. after localisation of a space,
This is analogous to complex projective space.
has infinite homotopy groups only in dimensions 4 and
, with respect to which it is a compact quaternion-Kähler symmetric space with positive curvature.
is the compact symplectic group.
, 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: where
[2] The one-dimensional projective space over
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.
Explicit expressions for coordinates for the 4-sphere can be found in the article on the Fubini–Study metric.
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.