Homogeneous space

Let X be a non-empty set and G a group. Then X is called a G-space if it is equipped with an action of G on X.[1] Note that automatically G acts by automorphisms (bijections) on the set. If X in addition belongs to some category, then the elements of G are assumed to act as automorphisms in the same category. That is, the maps on X coming from elements of G preserve the structure associated with the category (for example, if X is an object in Diff then the action is required to be by diffeomorphisms). A homogeneous space is a G-space on which G acts transitively.

Succinctly, if X is an object of the category C, then the structure of a G-space is a homomorphism:

${\displaystyle \rho :G\to \mathrm {Aut} _{\mathbf {C} }(X)}$

into the group of automorphisms of the object X in the category C. The pair (X, ρ) defines a homogeneous space provided ρ(G) is a transitive group of symmetries of the underlying set of X.

Examples

For example, if X is a topological space, then group elements are assumed to act as homeomorphisms on X. The structure of a G-space is a group homomorphism ρ : G → Homeo(X) into the homeomorphism group of X.

Similarly, if X is a differentiable manifold, then the group elements are diffeomorphisms. The structure of a G-space is a group homomorphism ρ : G → Diffeo(X) into the diffeomorphism group of X.

Concrete examples include:

Isometry groups

• Positive curvature:
  1. Sphere (orthogonal group): Sⁿ⁻¹ ≅ O(n) / O(n−1). This is true because of the following observations: First, Sⁿ⁻¹ is the set of vectors in Rⁿ with norm 1. If we consider one of these vectors as a base vector, then any other vector can be constructed using an orthogonal transformation. If we consider the span of this vector as a one dimensional subspace of Rⁿ, then the complement is an (n − 1)-dimensional vector space that is invariant under an orthogonal transformation from O(n − 1). This shows us why we can construct Sⁿ⁻¹ as a homogeneous space.
  2. Oriented sphere (
     special orthogonal group
     ): Sⁿ⁻¹ ≅ SO(n) / SO(n − 1)
  3. Projective space (projective orthogonal group): Pⁿ⁻¹ ≅ PO(n) / PO(n − 1)
• Flat (zero curvature):
  1. Euclidean space (Euclidean group, point stabilizer is orthogonal group): Aⁿ ≅ E(n) / O(n)
• Negative curvature:
  1. Hyperbolic space (
     orthochronous Lorentz group, point stabilizer orthogonal group, corresponding to hyperboloid model
     ): Hⁿ ≅ O⁺(1, n) / O(n)
  2. Oriented hyperbolic space: SO⁺(1, n) / SO(n)
  3. Anti-de Sitter space: AdS_n+1 = O(2, n) / O(1, n)

Others

• Affine space over field K (for affine group, point stabilizer general linear group): Aⁿ = Aff(n, K) / GL(n, K).
• Grassmannian: Gr(r, n) = O(n) / (O(r) × O(n − r))
• Topological vector spaces (in the sense of topology)

Geometry

From the point of view of the Erlangen program, one may understand that "all points are the same", in the geometry of X. This was true of essentially all geometries proposed before Riemannian geometry, in the middle of the nineteenth century.

Thus, for example, Euclidean space, affine space and projective space are all in natural ways homogeneous spaces for their respective symmetry groups. The same is true of the models found of non-Euclidean geometry of constant curvature, such as hyperbolic space.

A further classical example is the space of lines in projective space of three dimensions (equivalently, the space of two-dimensional subspaces of a four-dimensional

For example, if H is the identity subgroup {e}, then X is the G-torsor, which explains why G-torsors are often described intuitively as "G with forgotten identity".

In general, a different choice of origin o will lead to a quotient of G by a different subgroup H_o′ that is related to H_o by an inner automorphism of G. Specifically,

$${\displaystyle H_{o'}=gH_{o}g^{-1}}$$

(1)

where g is any element of G for which go = o′. Note that the inner automorphism (1) does not depend on which such g is selected; it depends only on g modulo H_o.

If the action of G on X is

For example, in the line geometry case, we can identify H as a 12-dimensional subgroup of the 16-dimensional general linear group, GL(4), defined by conditions on the matrix entries

h₁₃ = h₁₄ = h₂₃ = h₂₄ = 0,

by looking for the stabilizer of the subspace spanned by the first two standard basis vectors. That shows that X has dimension 4.

Since the homogeneous coordinates given by the minors are 6 in number, this means that the latter are not independent of each other. In fact, a single quadratic relation holds between the six minors, as was known to nineteenth-century geometers.

This example was the first known example of a Grassmannian, other than a projective space. There are many further homogeneous spaces of the classical linear groups in common use in mathematics.

Prehomogeneous vector spaces

The idea of a prehomogeneous vector space was introduced by Mikio Sato.

It is a finite-dimensional

Given the Poincaré group G and its subgroup the Lorentz group H, the space of cosets G / H is the spacetime algebra Minkowski space.^[2]

• Erlangen program
• Klein geometry
• Heap (mathematics)
• Homogeneous variety

Notes