What is 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 . {\displaystyle \mathbb {H}.} Quaternionic projective space of dimension n is usually denoted by

and is a closed manifold of dimension 4n. It is a homogeneous space for a Lie group action, in more than one way. The quaternionic projective line H P 1 {\displaystyle \mathbb {HP} ^{1}} is homeomorphic to the 4-sphere.