# 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 **H**. Quaternionic projective space of dimension *n* is usually denoted by

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

where the 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 of by the action of , the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside one may also regard as the orbit space of by the action of , the group of unit quaternions.^{[1]} The sphere then becomes a principal Sp(1)-bundle over :

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

There is also a construction of by means of two-dimensional complex subspaces of , meaning that lies inside a complex Grassmannian.

## 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 is the classifying space BS^{3}. The homotopy groups of are given by . 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 if and if . It follows that rationally, i.e. after localisation of a space, is an Eilenberg–Maclane space . That is . (cf. the example K(Z,2)). See rational homotopy theory.

## Quaternionic projective plane

The 8-dimensional 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.

## References

- ↑ Gregory L. Naber,
*Topology, geometry, and gauge fields: foundations*(1997), p. 50.

## Further reading

- V. I. Arnol'd,
*Relatives of the Quotient of the Complex Projective Plane by the Complex Conjugation*, Tr. Mat. Inst. Steklova, 1999, Volume 224, Pages 56–67. Treats the analogue of the result mentioned for quaternionic projective space and the 13-sphere.