The quaternionic projective space $\mathbb{H}P^n$ is the space of right (or left) quaternion lines through the origin in $\mathbb{H}^{n+1}$, hence the space of equivalence classes $[q_1, \cdots, q_{n+1}]$ of (n+1)-tuples of quaternions excluding zero, under the equivalence relation given by right (or left) multiplication with non-zero quaternions

$\mathbb{H}P^n
\;\coloneqq\;
\big\{
[q_1, \cdots, q_{n+1}]
\big\}
\;\coloneqq\;
\Big(
\big\{
(q_1, \cdots, q_{n+1})
\big\}
\setminus
\{(0, \cdots, 0)\}
\Big)
/_{
(q_1, \cdots, q_{n+1})
\sim
(q_1 q, \cdots, q_{n+1} q)
\vert
q \neq 0
}$

As any Grassmannian, quaternion projective space is canonically a coset space, in this case of the quaternion unitary group $Sp(n+1)$ by the central product group Sp(n).Sp(1):

(1)$\mathbb{H}P^n
\;\simeq\;
\frac{
Sp(n+1)
}{
Sp(n)\cdot Sp(1)
}$

By the coset space-realization (1), quaternion projective space is naturally a quaternion-Kähler manifold which is also a symmetric space. As such it is an example of a Wolf space.

- Pierre Conner, Edwin Floyd, Section I.4 of:
*The Relation of Cobordism to K-Theories*, Lecture Notes in Mathematics**28**Springer 1966 (doi:10.1007/BFb0071091, MR216511)

See also

- Wikipedia,
*Quaternionic projective space*

M-theory on the 8-manifold$\;$ HP2, hence on a quaternion-Kähler manifold of dimension 8 with holonomy Sp(2).Sp(1), is considered in

- Michael Atiyah, Edward Witten, p. 75 onwards in
*$M$-Theory dynamics on a manifold of $G_2$-holonomy*, Adv. Theor. Math. Phys. 6 (2001) (arXiv:hep-th/0107177)

and argued to be dual to M-theory on G2-manifolds in three different ways, which in turn is argued to lead to a a possible proof of confinement in the resulting 4d effective field theory (see there for more).

Last revised on February 6, 2021 at 23:59:15. See the history of this page for a list of all contributions to it.