nLab quaternionic projective space

Contents

Contents

Idea

The quaternionic projective space P n\mathbb{H}P^n is the space of right (or left) quaternion lines through the origin in n+1\mathbb{H}^{n+1}, hence the space of equivalence classes [q 1,,q n+1][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

P n{[q 1,,q n+1]}({(q 1,,q n+1)}{(0,,0)})/ (q 1,,q n+1)(q 1q,,q n+1q)|q0 \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 }

Properties

General

Proposition

Every continuous map P nP n\mathbb{H}P^n\rightarrow\mathbb{H}P^n for n2n \geq 2 has a fixed point. This does not hold for n=1n = 1 as in this case P 1S 4\mathbb{H}P^1\cong S^4 and the antipodal map S 4S 4,xxS^4\rightarrow S^4, x\mapsto -x does not have a fixed point.

(Hatcher 02, page 180)

Homology

Proposition

(homology of quaternionic projective space)

The ordinary homology groups of quaternionic projective space P n\mathbb{R}P^n can be calculated using its CW structure and are given by

(1)H k(P n)={ | k=0,4,,4(n1),4n 1 | otherwise H^k \big( \mathbb{H}P^n \big) \;=\; \left\{ \array{ \mathbb{Z} &\vert& \; k = 0, 4,\ldots, 4(n-1),4n \\ 1 &\vert& otherwise } \right.

As a coset space

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

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

As a quaternion-Kähler symmetric space (Wolf space)

By the coset space-realization (2), 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.

References

General

See also

In string theory

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

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 5, 2024 at 00:51:16. See the history of this page for a list of all contributions to it.