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. Quaternionic projective space of dimension n is usually denoted by Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, real projective space, or RPn is the projective space of lines in Rn+1. ... In mathematics, complex projective space, or CPn, is the projective space of (complex) lines in Cn+1. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...

HPⁿ

and is a closed manifold of (real) dimension 4n. It is a homogeneous space for a Lie group action, in more than one way. In mathematics, a closed manifold, or compact manifold, is a manifold that is compact as a topological space. ... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...

Its direct construction is as a special case of the projective space over a division algebra. The homogeneous coordinates of a point can be written In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius, allow affine transformations to be easily represented by a matrix. ...

[q₀:q₁: ... :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

[cq₀:cq₁: ... :cq_n].

In the language of group actions, HPⁿ is the orbit space of Hⁿ⁺¹ by the action of H*, the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside Hⁿ⁺¹ one may also regard HPⁿ as the orbit space of S⁴ⁿ⁺³ by the action of Sp(1), the group of unit quaternions. The sphere S⁴ⁿ⁺³ the becomes a principal Sp(1)-bundle over HPⁿ: In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves...

There is also a construction of HPⁿ by means of two-dimensional complex subspaces of C²ⁿ, meaning that HPⁿ lies inside a complex Grassmannian. In mathematics, a Grassmannian is the space of all k-dimensional subspaces of an n-dimensional vector space V, often denoted Gk(V) or simply Gk,n. ...

Quaternionic projective plane

The 8-dimensional HP² 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

HPⁿ/U(1)

may be taken, writing U(1) for the circle group. It has been shown that this quotient is the 7-sphere, a result of Vladimir Arnold from 1996, later rediscovered by Edward Witten and Michael Atiyah. In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices, with the group operation that of matrix multiplication. ... In mathematics, the circle group, denoted by T (or in blackboard bold by ), is the multiplicative group of all complex numbers with absolute value 1. ... A sphere is a perfectly symmetrical geometrical object. ... Vladimir I. Arnold (Moscow, December 2001). ... Edward Witten (born August 26, 1951) is an American mathematical physicist, Fields Medalist, and professor at the Institute for Advanced Study. ... Sir Michael Francis Atiyah, OM, FRS (born 22 April 1929) is a mathematician who was born in London. ...