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

HPⁿ

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

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

[q₀:q₁: ... :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

[cq₀:cq₁: ... :cq_n].

In the language of group actions, :HPⁿ is the orbit space of

Hⁿ⁺¹

by the action of H*, the non-zero quaternions.

There is also a construction of HPⁿ by means of two-dimensional complex subspaces of C²ⁿ, meaning that HPⁿ lies inside a complex Grassmannian.