While learning about the topology (particularly the homotopy) of adjoint and coadjoint orbits in Lie theory, I've stumbled across component groups. However I'm struggling to understand the significance of these objects.
I'd like to understand:
- How should I think about component groups intuitively?
- Why and how are component groups important?
- What is a nice, concrete example to help me understand component groups?
Suppose we have a Lie group $G$ and let $G^0$ denote the connected component of $G$ containing the identity. We can then define the component group to be $$ \mathcal{A} \left( G \right) = G / G^0. $$ It can be shown that $G^0$ is normal in $G$, so this quotient inherits a natural group structure.
By quotienting out the identity component $G^0$, we are identifying all elements $x \in G^0$ with the identity $1 \in G^0$. I can understand that if we quotiented the identity path-component, then the resulting space would be useful in studying the fundamental homotopy group. However that is not what we have done here. So why is this construction still useful?
Consider $O(3,\mathbb{R})$ (the usual othogonal group, linear isometries of $\mathbb{R}^3$ with the usual metric). This has two components, $SO(3)$, the group of orientation-preserving determinant 1 elements (rotations) and $-I\cdot SO(3)$, the orientation reversing or determinant $-1$ elements (rotation and a reflection). So $G/G^0$ is the group of order 2.
In any case, all the components are the "same" (torsors for the connected component of the identity), and the only interesting part for the purposes of homotopy is choosing one of these (choosing a base point), the canonical base point being the identity.
$G/G^0$ is some extra data (probably some finite or discrete group) that tells you how to build $G$ from $G^0$, interesting algebraically perhaps, but not for the purposes of homotopy.