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
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. 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.
The space , defined as the union of all finite 's under inclusion, is the classifying space BS3. 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
In general, has a cell structure with one cell in each dimension which is a multiple of 4, up to . Accordingly, its cohomology ring is , where is a 4-dimensional generator. This is analogous to complex projective space. It also follows from rational homotopy theory that has infinite homotopy groups only in dimensions 4 and .
Quaternionic projective space can be represented as the coset space
where is the compact symplectic group.
where is the generator of and is its reduction mod 2.
Quaternionic projective line
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. 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
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
- Naber, Gregory L. (2011) . "Physical and Geometrical Motivation". Topology, Geometry and Gauge fields. Texts in Applied Mathematics. 25. Springer. p. 50. doi:10.1007/978-1-4419-7254-5_0. ISBN 978-1-4419-7254-5.
- Szczarba, R.H. (1964). "On tangent bundles of fibre spaces and quotient spaces" (PDF). American Journal of Mathematics. 86 (4): 685–697. doi:10.2307/2373152. JSTOR 2373152.
- Arnol'd, V.I. (1999). "Relatives of the Quotient of the Complex Projective Plane by the Complex Conjugation". Tr. Mat. Inst. Steklova. 224: 56–6. CiteSeerX 10.1.1.50.6421. Treats the analogue of the result mentioned for quaternionic projective space and the 13-sphere.
- Gormley, P.G. (1947), "Stereographic projection and the linear fractional group of transformations of quaternions", Proceedings of the Royal Irish Academy, Section A, 51: 67–85, JSTOR 20488472