Let’s look back over what we’ve done.
After laying down some definitions on reflections, we defined a root system as a collection of vectors with certain properties. Specifically, each vector is a point in a vector space, and it also gives us a reflection of the same vector space. Essentially, a root system is a finite collection of such vectors and corresponding reflections so that the reflections shuffle the vectors among each other. Our project was to classify these configurations.
The flip side of seeing a root system as a collection of vectors is seeing it as a collection of reflections, and these reflections generate a group of transformations called the Weyl group of the root system. It’s one of the most useful tools we have at our disposal through the rest of the project.
To get a perspective on the classification, we defined the category of root systems. In particular, this leads us to the idea of decomposing a root system into irreducible root systems. If we can classify these pieces, any other root system will be built from them.
Like a basis of a vector space, a base of a vector space contains enough information to reconstruct the whole root system. Further, any two bases for a given root system look essentially the same, and the Weyl group shuffles them around. So really what we need to classify are the irreducible bases; for each such base there will be exactly one irreducible root system.
Finally, we could start the real work of classification: a list of the Dynkin diagrams that might arise from root systems. And then we could actually construct root systems that gave rise to each of these examples.
As a little followup, we could look back at the category of root systems and use the Dynkin diagrams and Weyl groups to completely describe the automorphism group of any root system.
Root systems come up in a number of interesting contexts. I’ll eventually be talking about them as they relate to Lie algebras, but (as we’ve just seen) they can be introduced and discussed as a self-motivated, standalone topic in geometry.