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

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):

(1)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 (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.

References

General

See also

In string theory

M-theory on the 8-manifoldP 2\mathbb{H}P^2, 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).

Created on April 13, 2019 at 06:30:33. See the history of this page for a list of all contributions to it.