Overview and definition.
Let be a group acting transitively on a set . In other words, we consider a homomorphism where the latter denotes the group of all bijections of . If we consider as being, in some sense, the automorphisms of , the transitivity assumption means that it is impossible to distinguish a particular element of from any another element. Since the elements of are indistinguishable, we call a homogeneous space. Indeed, the concept of a homogeneous space, is logically equivalent to the concept of a transitive group action.
Action on cosets.
The mapping is a bijection. The corresponding mapping is a group homomorphism, specifying a transitive group action of on .
N.B. In geometric applications, the want the homogeneous space to have some extra structure, like a topology or a differential structure. Correspondingly, the group of automorphisms is either a continuous group or a Lie group. In order for the quotient space to have a Hausdorff topology, we need to assume that the subgroup is closed in .
The isotropy subgroup and the basepoint identification.
Let be a homogeneous space. For , the subgroup
The above mapping is a well-defined bijection.
To show that is well defined, let be members of the same left coset, i.e. there exists an such that . Consequently
as desired. The mapping is onto because the action of on is assumed to be transitive. To show that is one-to-one, consider two cosets such that . It follows that fixes , and hence is an element of . Therefore and are the same coset.
The homogeneous space as a quotient.
We have that
for all .
To prove this, let be given, and note that
The latter coset corresponds under to the point , as desired.
Finally, let us note that identifies the point with the coset of the identity element , that is to say, with the subgroup itself. For this reason, the point is often called the basepoint of the identification .
The choice of basepoint.
Next, we consider the effect of the choice of basepoint on the quotient structure of a homogeneous space. Let be a homogeneous space.
The set of all isotropy subgroups forms a single conjugacy class of subgroups in .
To show this, let be given. By the transitivity of the action we may choose a such that . Hence, for all satisfying , we have
Similarly, for all we have that fixes . Therefore,
or what is equivalent, for all and we have
Since we can identify a homogeneous space with for every possible , it stands to reason that there exist equivariant bijections between the different . To describe these, let be conjugate subgroups with
for some fixed . Let us set
and let denote the identity coset , and the coset . What is the subgroup of that fixes ? In other words, what are all the such that
or what is equivalent, all such that
The collection of all such is precisely the subgroup . Hence, is the desired equivariant bijection. This is a well defined mapping from the set of -cosets to the set of -cosets, with action given by
Let and denote the corresponding coset -actions.
For all we have that
|Date of creation||2013-03-22 13:28:07|
|Last modified on||2013-03-22 13:28:07|
|Last modified by||rmilson (146)|
|Defines||action on cosets|