In

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

,

**quaternionic projective space** is an extension of the ideas of

real projective spaceIn mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...

and

complex projective spaceIn mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...

, to the case where coordinates lie in the ring of

quaternionIn mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

s

**H**. Quaternionic projective space of dimension

*n* is usually denoted by

- HP
^{}n

*
*

and is a closed manifoldIn mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....

of (real) dimension 4n

*. It is a homogeneous space*In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...

for a Lie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

action, in more than one way.

## In coordinates

Its direct construction is as a special case of the projective space over a division algebra. The homogeneous coordinatesIn mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...

of a point can be written

*
**[*q_{0}:q_{1}: ... :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*

*
**[*cq_{0}:cq_{1}: ... :cq_{}n*].*

*
*

In the language of group actionIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

s, HP^{n} is the orbit space of

**H**^{n+1}-(0, ..., 0) by the action of

**H***, the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside

**H**^{n+1} one may also regard

**H***P*^{}n* as the orbit space of *S^{4}n*+3 by the action of Sp(1), the group of unit quaternions. The sphere *S^{4}n*+3 then becomes a principal Sp(1)-bundle*In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...

over *H**P*^{n}:

There is also a construction of

**H***P*^{}n* by means of two-dimensional complex subspaces of **C*^{2}**n***, meaning that **H**P*^{n} lies inside a complex

GrassmannianIn mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...

.

## Projective line

The one-dimensional projective space over

**H** is called the "projective line" in generalization of the complex projective line. For example, it was used (implicitly) in 1947 by P. G. Gormley to extend the Möbius group to the quaternion context with "linear fractional transformations". See

inversive ring geometryIn mathematics, inversive ring geometry is the extension of the concepts of projective line, homogeneous coordinates, projective transformations, and cross-ratio to the context of associative rings, concepts usually built upon rings that happen to be fields....

for the uses of the projective line of the arbitrary

ringIn mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

.

From the topological point of view the quaternionic projective line is the 4-sphere, and in fact these are diffeomorphic manifolds. The fibration mentioned previously is from the 7-sphere, and is an example of a Hopf fibration.

## Infinite-dimensional quaternionic projective space

The space

is the

classifying spaceIn mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...

BS

^{3}; and, rationally (i.e. after localisation of a space it is an Eilenberg–Maclane space K(Z,4) (cf. the example K(Z,2)). See

rational homotopy theoryIn mathematics, rational homotopy theory is the study of the rational homotopy type of a space, which means roughly that one ignores all torsion in the homotopy groups...

.

## Quaternionic projective plane

The 8-dimensional

**H***P*^{}2* has a circle action, by the group of complex scalars of absolute value 1 acting on the other side (so on the right, as the convention for the action of *c* above is on the left). Therefore the quotient manifold*

**
***H*P^{2}/*U*(1)

may be taken, writing U(1) for the

circle group. It has been shown that this quotient is the 7-

sphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

, a result of

Vladimir ArnoldVladimir Igorevich Arnold was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory,...

from 1996, later rediscovered by

Edward WittenEdward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....

and

Michael AtiyahSir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...

.

## Further reading

- V. I. Arnol'd,
*Relatives of the Quotient of the Complex Projective Plane by the Complex Conjugation*, Tr. Mat. Inst. Steklova, 1999, Volume 224, Pages 56–67. Treats the analogue of the result mentioned for quaternionic projective space and the 13-sphere.