Everything ends and it’s always sad, but everything begins again, too.

And that’s always happy.

Be happy.

— 12th Doctor, “Return of Doctor Mysterio”

Learning something new is always confusing for everyone, but it’s also exciting for everyone. What matters is if latter exceeds former or vice versa.

If you got some idea of what Group is in mathematics, it’s time to think about it’s applications. No, I’m not talking about how can we use the concept for buying 241 sandwitches. So-called “real life” is not our concern. We have to think how can we use group in the domain of mathematics. What would be some interesting new mathematical concept we can make with group?

One of them is what we call a Lie Group.

Definition up front, Lie Group is “A group with smooth manifold.”

Informally, Lie group is a group of symmetries where the symmetries are continuous. (Source: aimath.org)

Okay, that was a short and memorable definition. But what in the world is a “smooth structure? You might already have some idea with what smoothness is (is your stuff differntiable?). To move on, let’s first delare that smooth is equivalent of saying ”differentiable.” However, smooth manifold (or differentiable manifold) is something a bit abstract we cannot process in our brain in the first place.

Manifold

Manifold is a mathematical concept with quite a bit of abstractness, which is a topological space. When we imagine a snippit of manifold in our brain, we can think of a set of cartesian space. However, what makes it special is its locality. The manifold acts ‘Euclidically’ as a cartesian space, but it’s only when it is local enough. For example, sphere is also a manifold. We can project the small local area of the sphere on a Euclidean space. And one of the most popular sphere in our society is what we call the Earth. We can draw a world map on the flat paper and still use it without much trouble.

Therefore, manifold is a topological space that is locally euclidian. For an arbitary mathematical object to become a manifold, there are 4 requirements, thanks to Brian Keng’s blog. I numbered them so that we can easily identify each rules:

1. The mapping \(\varphi: U \rightarrow V\) is called a chart or coordinate system.
2. The set \(U\) is the domain or local coordinate neighbourhood of the chart.
3. The image of the point \(p \in U\), denoted by \(\varphi(p) \in \mathbb{R}^n\), is called the coordinates or local coordinates of \(p\) in the chart.
4. A set of charts, \(\{\varphi_\alpha | \alpha \in \mathbb{N}\}\), with domains \(U_\alpha\) is called the atlas¹ of M, if \(\bigcup\limits_{\alpha \in \mathbb{N}} U_\alpha = M.\)²

Now let’s decompose these rules so that we can swallow them up. While these look cryptic for some readers, this is not that difficult to interprete. Basically, these are simply a declaration of methods and elements in the manifold. Which means, we can think of this as a class of Object-Oriented Programming and its methods/variables inside, but with a bit more rigor.

It includes the method \(\varphi: U \rightarrow V\) is the chart function (method), also known as the coordinate system, which maps the arbitary point \(p \in U\) into another arbitary point in the nth-dimensional real space \(\mathbb{R}^n\) (That means, obviously, there can be 2-dimensional manifold as well, that is basically a straight line which might help some people’s understanding). And we can represent the mapped point like \(\varphi(p) \in \mathbb{R}^n\), then name the mapped point (output) as (local) coordinates of \(p\). That is what rule 1~3 tells us.

To help your understanding, let’s draw the map of Japan. We have one huge map of entire Japan with extreme accuracy. This map includes all the building-level information inside, and it is really huge when you completely unfold it. However, you also have 47 small maps of Japanese prefectures. And now we want to take a look at Shimogamo Yusui Villa (下鴨幽水荘) in Kyoto. (image source: twitter)

If your “one huge map” is precise enough, you will find Asakusa easily by looking for the bigger areas like in this order: Kansai -> Kyoto -> Shimogamo Yusui Villa.

When you’re traveling in Kyoto, it is more likely you have the smaller map of Just Kyoto, rather than a huge map which might be taller than you. Even though, when you want to calculate the travel time between your location in Kyoto and somewhere else in Hokkaido, you will have to look at the huge on as well. You might want to match your location in the huge map to that of the “local” Kyoto guidebook. You will naturally understand those two are literally same places, because it’s the place you’re standing right now under the merciless sunlight of Kyoto, with a bottle of sports drink.

The function \(\varphi\) is for this sort of job. This is the mapper with the type signature such as phi :: Point -> LocalCoord. Takes the global point of the argument and returns the corresponding local coordinate.

Although, here is something we should keep in mind: the mapping should be bijective (one-to-one). Which means, there should not be more than 2 same places on the prefecture map. You can’t find 2 or 3 city halls in the city map. This might look trivial and evident, but everything becomes tricky when we start to think about the map of the Earth. Since the Earth is round, if you expand your map a bit too much, you might end up with two Statue(s) of Liberty on it.

Lastly, Rule 4 tells us there is an Akashic Record of the space we’re talking about. In the Age of Discovery, each adventurers explored all the different places over the globe and drew a map. This corresponds to a chart of manifold. And to get the wider perspective of their adventures, they gathered their maps one place, and made a map book, also known as atlas. That’s the etymology of the atlas in our context.

Smooth

So, we got the grasp of idea regarding the manifold, this is the right moment to remind the loose definition of Lie Group.

Lie group (pronounced /liː/ LEE) is a group that is also a differentiable manifold. (Source: Wikipedia)

Good. We studied about manifold till this point, and Lie Group is a manifold. We have just one more step to reach our final destination: understanding the smooth manifold.

Now when we heard some magic words like smooth or differentiable, our intuition tells us it must be something continuous and not discrete, from the nuance of its words. But what is not discrete and continuous in the smooth manifold?