Homogeneous space

From Example Problems
Jump to navigation Jump to search

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. That is, there is a group action of G on X, respecting the given geometric structure of X, and making X into a single G-orbit. (It is assumed, therefore, that X isn't empty.) If X is simply called a homogeneous space without reference to a group, it is usually assumed that G is the group of all homeomorphisms from X to itself (the automorphism group of X), with the natural action of evaluation.

From the point of view of the Erlangen programme, 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. 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 vector space). It is simple linear algebra to show that GL4 acts transitively on those. We can parameterize them by line co-ordinates: these are the 2×2 minors of the 2×4 matrix with columns two basis vectors for the subspace. The geometry of the resulting homogeneous space is the line geometry of Julius Plücker.

In general, if X is a homogeneous space, and H is the stabilizer of some fixed x in X, the points of X correspond to the cosets G/H. We can assume that H is a closed subgroup of G, for a continuous action: when it is the identity subgroup {e}, we have a principal homogeneous space.

For example in the line geometry example we can identify H as a 12-dimensional subgroup of the 16-dimensional group GL4, 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 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.

The idea of a prehomogeneous vector space was introduced by Mikio Sato. It is a finite-dimensional vector space V with a group action of an algebraic group G, such that there is an orbit of G that is open for the Zariski topology (and so, dense). An example is GL1 acting on a one-dimensional space. The definition is more restrictive than it initially appears: such spaces have remarkable properties, and there is a classification.

See also