It might seem like were pretty far in the weeds by now but there’s a point to make here which is essential to my work. And we may as well introduce some of that noxious stuff — mathematical jargon. The collection of symmetries we’ve been talking about that can act in sequence or be done and then undone is usually called a group in mathematics. Maybe because it’s sort of like a group that gets together to play ball (though it might be with a line). By the way — not everyone in the group actually knows how to really play. There is always one character called the identity that just leaves the object as it is, moving nothing. This is still a symmetry though, in the same way 0 is a number, which is worth thinking about some.
An interesting question to ask is: where can points on the line go as the group bobbles it around? To simplify, how about where does a particular point, say zero, go? Where it ends up depends on the particular symmetry we apply, but considering all possibilities we see that zero can really go anywhere on the line. How? Well, if we want to take 0 to some value we actually have two options: either apply the translation that adds to every point, or apply the reflection through the value . Since the point 0 can go anywhere, we say that its orbit under the group is the whole line.
We’ve seen that we can think of the members (elements) of our group as constituting two copies of the line. On the first line, each point is a translation. On the second line, each point is a reflection (which can actually be realized as a translation and a reflection through 0). The fact that we have two ways to take 0 to, say, mirrors the fact that the group looks like two copies of the line.
So here’s a key idea: probably you would agree that the line is a natural enough geometric object. We need lines to get notions of distance and angle going in geometry! On the other hand, this group which is two copies of the line seems a bit funny and abstract. But it is still intimately attached to the line by keeping track of its symmetries. So what if there was a way to have both? That is, to have just the line in hand as an object but manifest in a way that also keeps track of the symmetry. This is what homogeneous spaces are for.
To explain how we get a homogeneous space here, I need to try to explain something in group theory called the orbit-stabilizer theorem. Now, I don’t really know how to do this without either bringing in a little notation or being extremely verbose (and likely confusing). But there is power in notation so let’s try that. We’ve observed that all reflections can be considered as reflections through zero together with a translation. Lets call the reflection through zero “flip,” and label translations by the value that is added to every point, . Then every element of our group is a pair of the form (, flip) or (, no flip) according to whether it is a reflection or a translation.
Notice that the reflection through zero (0, flip) “fixes” zero, that is leaves it put, and other than the identity, which can be thought of as (0, no flip), it is the only element that does this. In group theory terms, these two elements form a subgroup which consists of the elements that fix 0. The orbit-stabilizer theorem says that if we clump these two elements together and think of them as a single object, and then do the same for all of the pairs (, flip) and (, no flip), then the resulting collection will be the same as the orbit of the point 0. But this is the just the line again! This should make some sense; in our group, we had two copies of the line and all we’ve done is glued those copies together at points that line up.
This “clumping” of objects according to a subgroup is called taking a quotient in group theory, and a homogeneous space is really just a quotient. It’s useful because now the coordinates on the line now come from the coordinates on the group, which facilitates studying how the symmetries act, especially in more complicated examples. A lot of my work is studying how these quotient spaces break up and fit together in pieces when you restrict your symmetry set. We’ll try to come back to this in the next post.