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 H. Quaternionic projective space of dimension n is usually denoted by
Contents
- In coordinates
- Projective line
- Infinite dimensional quaternionic projective space
- Quaternionic projective plane
- References
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
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
where the
In the language of group actions,
This bundle is sometimes called a (generalized) Hopf fibration.
There is also a construction of
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 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.
Infinite-dimensional quaternionic projective space
The space
Quaternionic projective plane
The 8-dimensional
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.