homogeneous space

Overview and definition.

Let $G$ be a group acting transitively on a set $X$. In other words, we consider a homomorphism $\phi:G\to\operatorname{Perm}(X),$ where the latter denotes the group of all bijections of $X$. If we consider $G$ as being, in some sense, the automorphisms of $X$, the transitivity assumption means that it is impossible to distinguish a particular element of $X$ from any another element. Since the elements of $X$ are indistinguishable, we call $X$ a homogeneous space. Indeed, the concept of a homogeneous space, is logically equivalent to the concept of a transitive group action.

Action on cosets.

Let $G$ be a group, $H a subgroup, and let $G/H$ denote the set of left cosets, as above. For every $g\in G$ we consider the mapping $\psi_{H}(g):G/H\to G/H$ with action

 $aH\to gaH,\quad a\in G.$
Proposition 1

The mapping $\psi_{H}(g)$ is a bijection. The corresponding mapping $\psi_{H}:G\to\operatorname{Perm}(G/H)$ is a group homomorphism, specifying a transitive group action of $G$ on $G/H$.

Thus, $G/H$ has the natural structure of a homogeneous space. Indeed, we shall see that every homogeneous space $X$ is isomorphic to $G/H$, for some subgroup $H$.

N.B. In geometric applications, the want the homogeneous space $X$ 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 $X$ to have a Hausdorff topology, we need to assume that the subgroup $H$ is closed in $G$.

The isotropy subgroup and the basepoint identification.

Let $X$ be a homogeneous space. For $x\in X$, the subgroup

 $H_{x}=\{h\in G:hx=x\},$

consisting of all $G$-actions that fix $x$, is called the isotropy subgroup at the basepoint $x$. We identify the space of cosets $G/H_{x}$ with the homogeneous space by means of the mapping $\tau_{x}:G/H_{x}\to X$, defined by

 $\tau_{x}(aH_{x})=ax,\quad a\in G.$
Proposition 2

The above mapping is a well-defined bijection.

To show that $\tau_{x}$ is well defined, let $a,b\in G$ be members of the same left coset, i.e. there exists an $h\in H_{x}$ such that $b=ah$. Consequently

 $bx=a(hx)=ax,$

as desired. The mapping $\tau_{x}$ is onto because the action of $G$ on $X$ is assumed to be transitive. To show that $\tau_{x}$ is one-to-one, consider two cosets $aH_{x},bH_{x},\;a,b\in G$ such that $ax=bx$. It follows that $b^{-1}a$ fixes $x$, and hence is an element of $H_{x}$. Therefore $aH_{x}$ and $bH_{x}$ are the same coset.

The homogeneous space as a quotient.

Next, let us show that $\tau_{x}$ is equivariant relative to the action of $G$ on $X$ and the action of $G$ on the quotient $G/H_{x}$.

Proposition 3

We have that

 $\phi(g)\circ\tau_{x}=\tau_{x}\circ\psi_{H_{x}}(g)$

for all $g\in G$.

To prove this, let $g,a\in G$ be given, and note that

 $\psi_{H_{x}}(g)(aH_{x})=gaH_{x}.$

The latter coset corresponds under $\tau_{x}$ to the point $gax$, as desired.

Finally, let us note that $\tau_{x}$ identifies the point $x\in X$ with the coset of the identity element $eH_{x}$, that is to say, with the subgroup $H_{x}$ itself. For this reason, the point $x$ is often called the basepoint of the identification $\tau_{x}:G/H_{x}\to X$.

The choice of basepoint.

Next, we consider the effect of the choice of basepoint on the quotient structure of a homogeneous space. Let $X$ be a homogeneous space.

Proposition 4

The set of all isotropy subgroups $\{H_{x}:x\in X\}$ forms a single conjugacy class of subgroups in $G$.

To show this, let $x_{0},x_{1}\in X$ be given. By the transitivity of the action we may choose a $\hat{g}\in G$ such that $x_{1}=\hat{g}x_{0}$. Hence, for all $h\in G$ satisfying $hx_{0}=x_{0}$, we have

 $(\hat{g}h\hat{g}^{-1})x_{1}=\hat{g}(h(\hat{g}^{-1}x_{1}))=\hat{g}x_{0}=x_{1}.$

Similarly, for all $h\in H_{x_{1}}$ we have that $\hat{g}^{-1}h\hat{g}$ fixes $x_{0}$. Therefore,

 $\hat{g}(H_{x_{0}})\hat{g}^{-1}=H_{x_{1}};$

or what is equivalent, for all $x\in X$ and $g\in G$ we have

 $gH_{x}g^{-1}=H_{gx}.$

Equivariance.

Since we can identify a homogeneous space $X$ with $G/H_{x}$ for every possible $x\in X$, it stands to reason that there exist equivariant bijections between the different $G/H_{x}$. To describe these, let $H_{0},H_{1} be conjugate subgroups with

 $H_{1}=\hat{g}H_{0}\hat{g}^{-1}$

for some fixed $\hat{g}\in G$. Let us set

 $X=G/H_{0},$

and let $x_{0}$ denote the identity coset $H_{0}$, and $x_{1}$ the coset $\hat{g}H_{0}$. What is the subgroup of $G$ that fixes $x_{1}$? In other words, what are all the $h\in G$ such that

 $h\hat{g}H_{0}=\hat{g}H_{0},$

or what is equivalent, all $h\in G$ such that

 $\hat{g}^{-1}h\hat{g}\in H_{0}.$

The collection of all such $h$ is precisely the subgroup $H_{1}$. Hence, $\tau_{x_{1}}:G/H_{1}\to G/H_{0}$ is the desired equivariant bijection. This is a well defined mapping from the set of $H_{1}$-cosets to the set of $H_{0}$-cosets, with action given by

 $\tau_{x_{1}}(aH_{1})=a\hat{g}H_{0},\quad a\in G.$

Let $\psi_{0}:G\to\operatorname{Perm}(G/H_{0})$ and $\psi_{1}:G\to\operatorname{Perm}(G/H_{1})$ denote the corresponding coset $G$-actions.

Proposition 5

For all $g\in G$ we have that

 $\tau_{x_{1}}\circ\psi_{1}(g)=\psi_{0}(g)\circ\tau_{x_{1}}.$
Title homogeneous space HomogeneousSpace 2013-03-22 13:28:07 2013-03-22 13:28:07 rmilson (146) rmilson (146) 6 rmilson (146) Definition msc 20A05 action on cosets isotropy subgroup