Quaternionic projective space

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

:H"P""n"

and is a closed manifold of (real) dimension "4n". It is a homogeneous space for a Lie group action, in more than one way.

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

: ["q"0:"q"1: ... :"q""n"]

where the "q""i" 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

: ["cq"0:"cq"1: ... :"cq""n"] .

In the language of group actions, H"P""n" is the orbit space of H"n"+1 by the action of H*, the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside H"n"+1 one may also regard H"P""n" as the orbit space of "S"4"n"+3 by the action of Sp(1), the group of unit quaternions. The sphere "S"4"n"+3 then becomes a principal Sp(1)-bundle over H"P""n"::mathrm{Sp}(1) o S^{4n+3} o mathbb HP^n.

There is also a construction of H"P""n" by means of two-dimensional complex subspaces of C2"n", meaning that H"P""n" lies inside a complex Grassmannian.

Infinite-dimensional quaternionic projective space

The space mathbb{HP}^{infty} is BS3 and, rationally, K(Z,4) (cf. K(Z,2)). See rational homotopy theory. Please expand.

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 Mobius group to the quaternion context with "linear fractional transformations". See inversive ring geometry for the uses of the projective line of the arbitrary ring.

Quaternionic projective plane

The 8-dimensional H"P""2" 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

:H"P""n"/"U"(1)

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.


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