Summary of Core-ideas

It is often helpful to study a homogeneous space of a group.

We can start with a group and a subgroup to build the corresponding quotient space, which is homogeneous to the original group. Conversely, we can start with a space and use the stabilizer of some element of this space to build the subgroup that will lead to a quotient space, identifying (i.e., there exists an isomorphism between the two spaces) this space.

Concepts

When will we say a group action is transitive?

A group action \(\odot:G\times\mathcal{X}\rightarrow\mathcal{X}\) is transitive if \(\forall x, \tilde{x}\in\mathcal{X},\exists g\in G\), s.t. \(\tilde{x}=g\odot x\).

When will a space be called homogeneous space for the group acting on it?

  • the group is a Lie group
  • the group action is transitive

What is a semidirect-product?

Basically, it is a group.

It is constructed from two groups \(N\) and \(H\) with a group action \(\odot: H\times N\rightarrow N\). We denote the semidirect-product by \(N\rtimes H\).

The group product and inverse is defined as:

\[(n,h)\cdot(\tilde{n},\tilde{h}) = (n\cdot (h\odot\tilde{n}), h\cdot\tilde{h})\] \[(n,h)^{-1}=(h^{-1}\odot n^{-1}, h^{-1})\]

Note that, for simplicity, we use the same symbol \(\cdot\) for these two groups’ respective group product.

What is a coset?

For a group \(G\) and a subgroup of it \(H\), $$gH={g\cdot h h\in H}\(for any group element\)g\in G$$ is a coset.

What is quotient space?

Given a group \(G\) and its subgroup \(H\), the quotient space \(G/H\) denotes the collection of distinct cosets $${gH g\in G}$$.

Thus, elements in \(G/H\) are equivalence classes, where any \(g\neq\tilde{g}\) are in the same class if \(\exists h\in H\) s.t., \(g=\tilde{g}\cdot h\).

What is a stabilizer?

Suppose there is a group action \(\odot: G\times \mathcal{X}\rightarrow \mathcal{X}\), then the stabilizer (subgroup) of \(G\) w.r.t. \(x_0 \in \mathcal{X}\) is defined as: \(Stab_{G}(x_0)=\{g|g\odot x_0 = x_0 \}\)

What is affine group?

Groups constructed by \(\Re^{d}\rtimes H\) for a certain \(H\subseteq\text{GL}(\Re^{d})\), where \(\text{GL}(\Re^d)\) denotes the general linear transformations acting on \(\Re^d\). \(\text{GL}(\Re^d)\) consists of invertible matrices acting on \(\Re^d\), and \(H\) is commonly a subgroup of \(\text{GL}(\Re^d)\).

Examples and Interpretations

Any example of semi-product?

Consider \(SE(d)=(\Re^{d},+)\rtimes SO(d)\). Then the group product and inverse element can be expressed as:

\[(x, R)\cdot(\tilde{x},\tilde{R})=(x+R\tilde{x}, R\tilde{R})\] \[(x, R)^{-1}=(-R^{-1}x, R^{-1})\]

where \(x\in\Re^{d}\) and \(R\) is the corresponding rotation matrix of a specific angle.

Are cosets always groups?

No, a coset may not be a group.

Any example of quotient space?

Suppose \(G=\{0, 1, \ldots, 7\}\), and the group product is defined as \(g\cdot\tilde{g}=(g+\tilde{g})\text{ mod }8\).

Suppose \(H=\{0, 4\}\), then the quotient space \(G / H\) consists of \(\{0, 4\}, \{1, 5\}, \{2, 6\}, \{3, 7\}\).

What’s the relationship between \(G\) and its quotient space \(G/H\)?

Define the group action \(i \odot gH\) to be $${i\cdot j j\in gH}$$.

Then, for any \(gH\) and \(\tilde{g}H\), \(\exists i\in G\) s.t., \(i\odot gH = \tilde{g}H\), because, to let \(i\cdot g\cdot h = \tilde{g}\cdot h\), we could just let \(i=\tilde{g} \cdot g^{-1}\).

Thus, \(G/H\) is a homogeneous space of \(G\).

How to interpret affine space from the perspective of quotient space?

Taking SE(2)=\(\Re^2 \rtimes S^1\) as an example. The group product, according to the definition of semidirect product, is \(g_1 \cdot g_2 =(x_1, \theta_1) \cdot (x_2, \theta_2) = (x_1 + R_{\theta_1}x_2, \theta_1 + \theta_2)\). Obviously, $${(0, \theta) \theta \in S^1 }\(is a subgroup, where\)(0, 0)\(is the group product identity, and closure is preserved. Then each equivalence class is in the form of\){ (x, \theta) \theta \in S^1 }$$.

Why can the representation of a semidirect product be decomposed into the function composition of their respective representations?