As we deal with algebras of sets, we’ll be wanting to take products of these structures. But it’s not as simple as it might seem at first. We won’t focus, yet, on the categorical perspective, and will return to that somewhat later.
Okay, so what’s the problem? Well, say we have sets and , and algebras of subsets and . We want to take the product set and come up with an algebra of sets . It’s sensible to expect that if we have and , we should have . Unfortunately, the collection of such products is not, itself, an algebra of sets!
So here’s where our method of generating an algebra of sets comes in. In fact, let’s generalize the setup a bit. Let’s say we’ve got which generates as the collection of finite disjoint unions of sets in , and let be a similar collection. Of course, since the algebras and are themselves closed under finite disjoint unions, we could just take and , but we could also have a more general situation.
Now we can define to be the collection of products of sets and , and we define as the set of finite disjoint unions of sets in . I say that satisfies the criteria we set out yesterday, and thus is an algebra of subsets of .
First off, is in both and , and so is in . On the other hand, and , so is in . That takes care of the first condition.
Next, is closed under pairwise intersections? Let and be sets in A point is in the first of these sets if and ; it’s in the second if and . Thus to be in both, we must have and . That is,
Since and are themselves closed under intersections, this set is in .
Finally, can we write as a finite disjoint union of sets in ? A point is in this set if it misses in the first coordinate — and — or if it does hit but misses in the second coordinate — and . That is:
Now , and so it can be written as a finite disjoint union of sets in ; thus can be written as a finite disjoint union of sets in . Similarly, we see that can be written as a finite disjoint union of sets in . And no set from the first collection can overlap any set in the second collection, since they’re separated by the first coordinate being contained in or not. Thus we’ve written the difference as a finite disjoint union of sets in , and so .
Therefore, satisfies our conditions, and is the algebra of sets it generates.
We might not always want to lay out an entire algebra of sets in one go. Sometimes we can get away with a smaller collection that tells us everything we need to know.
Suppose that is a subset of — a collection of subsets of — and define to be the collection of finite disjoint unions of subsets in . If we impose the following three conditions on :
- The empty set and the whole space are both in .
- If and are in , then so is their intersection .
- If and are in , then their difference is in
then is an algebra of sets.
If , then , and so contains and . We can also find , since .
Let’s take and to be two sets in , written as finite disjoint unions of sets in . Then their intersection is
Each of the is in , as an intersection of two sets in , and no two of them can intersect. Thus finite intersections of sets in are again in .
If , then . Since each of the are in , their (finite) intersection must be as well, and is closed under complements.
And so we can find that if and are in , then and are both in , and is thus an algebra of sets.
Okay, now that root systems are behind us, I’m going to pick back up with some analysis. Specifically, more measure theory. But it’s not going to look like the real analysis we’ve done before until we get some abstract basics down.
We take some set , which we want to ultimately consider as a sort of space so that we can measure parts of it. We’ve seen before that the power set — the set of all the subsets of — is an orthocomplemented lattice. That is, we can take meets (intersections) , joins (unions) and complements of subsets of , and these satisfy all the usual relations. More generally, we can use these operations to construct differences .
Now, an algebra of subsets of will be just a sublattice of which contains both the bottom and top of : the empty subset and the whole set . The usual definition is that if it contains and , then it contains both the union and the difference , along with and . But from this we can get complements — — and DeMorgan’s laws give us intersections — .
It’s important to note here that these operations let us define finite unions and intersections, just by iteration. But finite operations like this are just algebra. What makes analysis analysis is limits. And so we want to add an “infinite” operation.
Let’s say we have a countably infinite collection of subsets, . Then we define the countable union as a limit
We could also just say that the countable union consists of all points in any of the , but it will be useful to explicitly think of this as a process: Starting with we add in , then , and so on. If for some , then by the time we reach the th step we’ve folded into the growing union. The countable union is the limit of this process.
This viewpoint also brings us into contact with the category-theoretic notion of a colimit (feel free to ignore this if you’re category-phobic). Indeed, if we define and
then clearly we have an inclusion mapping for every natural number . That is, we have a functor from the natural numbers as an order category to the power set considered as one. And the colimit of this functor is the countable union.
So, let’s say we have an algebra of subsets of and add the assumption that is closed under such countable unions. In this case, we say that is a “-algebra”. We can extend DeMorgan’s laws to show that a -algebra will be closed under countable intersections as well as countable unions.
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.
Finally, we’re able to determine the automorphism group of our root systems. That is, given an object in the category of root systems, the morphisms from that root system back to itself (as usual) form a group, and it’s interesting to study the structure of this group.
First of all, right when we first talked about the category of root systems, we saw that the Weyl group of is a normal subgroup of . This will give us most of the structure we need, but there may be automorphisms of that don’t come from actions of the Weyl group.
So fix a base of , and consider the collection of automorphisms which send back to itself. We’ve shown that the action of on bases of is simply transitive, which means that if comes from the Weyl group, then can only be the identity transformation. That is, as subgroups of .
On the other hand, given an arbitrary automorphism , it sends to some other base . We can find a sending back to . And so ; it’s an automorphism sending to itself. That is, ; any automorphism can be written (not necessarily uniquely) as the composition of one from and one from . Therefore we can write the automorphism group as the semidirect product:
All that remains, then, is to determine the structure of . But each shuffles around the roots in , and these roots correspond to the vertices of the Dynkin diagram of the root system. And for to be an automorphism of , it must preserve the Cartan integers, and thus the numbers of edges between any pair of vertices in the Dynkin diagram. That is, must be a transformation of the Dynkin diagram of back to itself, and the reverse is also true.
So we can determine just by looking at the Dynkin diagram! Let’s see what this looks like for the connected diagrams in the classification theorem, since disconnected diagrams just add transformations that shuffle isomorphic pieces.
Any diagram with a multiple edge — , , and the and series — has only the trivial symmetry. Indeed, the multiple edge has a direction, and it must be sent back to itself with the same direction. It’s easy to see that this specifies where every other part of the diagram must go.
The diagram is a single vertex, and has no nontrivial symmetries either. But the diagram for can be flipped end-over-end. We thus find that for all these diagrams. The diagram can also be flipped end-over-end, leaving the one “side” vertex fixed, and we again find , but and have no nontrivial symmetries.
There is a symmetry of the diagram that swaps the two “tails”, so for . For , something entirely more interesting happens. Now the “body” of the diagram also has length , and we can shuffle it around just like the “tails”. And so for we find — the group of permutations of these three vertices. This “triality” shows up in all sorts of interesting applications that connect back to Dynkin diagrams and root systems.
Today we construct the last of our root systems, following our setup. These correspond to the Dynkin diagrams , , and . But there are transformations of Dynkin diagrams that send into , and on into . Thus all we really have to construct is , and then cut off the right simple roots in order to give , and then .
We start similarly to our construction of the root system; take the eight-dimensional space with the integer-coefficient lattice , and then build up the set of half-integer coefficient vectors
Starting from lattice , we can write a generic lattice vector as
and we let be the collection of lattice vectors so that the sum of the coefficients is even. This is well-defined even though the coefficients aren’t unique, because the only redundancy is that we can take from and add to each of the other eight coefficients, which preserves the total parity of all the coefficients.
Now let consist of those vectors with . The explicit description is similar to that from the root system. From , we get the vectors , but not the vectors because these don’t make it into . From we get some vectors of the form
Starting with the choice of all minus signs, this vector is not in because and all the other coefficients are . To flip a sign, we add , which flips the total parity of the coefficients. Thus the vectors of this form that make it into are exactly those with an odd number of minus signs.
We need to verify that for all and in (technically we should have done this yesterday for , but here it is. If both and come from , this is clear since all their coefficients are integers. If and , then the inner product is the sum of the th and th coefficients of , but with possibly flipped signs. No matter how we choose and , the resulting inner product is either , , or . Finally, if both and are chosen from , then each one is plus an odd number of the , which we write as and , respectively. Thus the inner product is
The first term here is , and the last term is also an integer because the coefficients of and are all integers. The middle two terms are each a sum of an odd number of , and so each of them is a half-integer. The whole inner product then is an integer, as we need.
What explicit base should we pick? We start out as we’ve did for with , , and so on up to . These provide six of our eight vertices, and the last two of them are perfect for cutting off later to make the and root systems. We also throw in , like we did for the series. This provides us with the triple vertex in the Dynkin diagram.
We need one more vertex off to the left. It should be orthogonal to every one of the simple roots we’ve chosen so far except for , with which it should have the inner product . It should also be a half-integer root, so that we can get access to the rest of them. For this purpose, we choose the root . Establishing that the reflection with respect to this vector preserves the lattice — and thus the root system — proceeds as in the case.
The Weyl group of is again the group of symmetries of a polytope. In this case, it turns out that the vectors in are exactly the vertices of a regular eight-dimensional polytope inscribed in the sphere of radius , and the Weyl group of is exactly the group of symmetries of this polyhedron! Notice that this is actually something interesting; in the case the roots formed the vertices of a hexagon, but the Weyl group wasn’t the whole group of symmetries of the hexagon. This is related to the fact that the diagram possesses a symmetry that flips it end-over-end, and we will explore this behavior further.
The Weyl groups of and are also the symmetries of seven- and six-dimensional polytopes, respectively, but these aren’t quite so nicely apparent from their root systems.
As the most intricate (in a sense) of these root systems, has inspired quite a lot of study and effort to visualize its structure. I’ll leave you with an animation I found on Garrett Lisi’s notewiki, Deferential Geometry (with the help of Sarah Kavassalis).
Today we construct the root system starting from our setup.
As we might see, this root system lives in four-dimensional space, and so we start with this space and its integer-component lattice . However, we now take another copy of and push it off by the vector . This set consists of all vectors each of whose components is half an odd integer (a “half-integer” for short). Together with , we get a new lattice consisting of vectors whose components are either all integers or all half-integers. Within this lattice , we let consist of those vectors of squared-length or : or ; we want to describe these vectors explicitly.
When we constructed the and series, we saw that the vectors of squared-length and in are those of the form (squared-length ) and of the form for (squared-length ). But what about the vectors in ? We definitely have — with squared-length — but can we have any others? The next longest vector in will have one component and the rest , but this has squared-length and won’t fit into ! We thus have twenty-four long roots of squared-length and twenty-four short roots of squared-length .
Now, of course we need an explicit base , and we can guess from the diagram that two must be long and two must be short. In fact, in a similar way to the root system, we start by picking and as two long roots, along with as one short root. Indeed, we can see a transformation of Dynkin diagrams sending into , and sending the specified base of to these three vectors.
But we need another short root which will both give a component in the direction of and will give us access to . Further, it should be orthogonal to both and , and should have a Cartan integer of with in either order. For this purpose, we pick , which then gives us the last vertex of the Dynkin diagram.
Does the reflection with respect to this last vector preserve the root system, though? What is its effect on vectors in ? We calculate
Now the sum is always an integer, whether the components of are integers or half-integers. If the sum is even, then we are changing each component of by an integer, which sends and back to themselves. If the sum is off, then we are changing each component of by a half-integer, which swaps and . In either case, the lattice is sent back to itself, and so this reflection fixes .
Like we say for it’s difficult to understand the Weyl group of in terms of its action on the components of . However, also like , we can understand it geometrically. But instead of a hexagon, now the long and short roots each make up a four-dimensional polytope called the “24-cell”. It’s a shape with 24 vertices, 96 edges, 96 equilateral triangular faces, and 24 three-dimensional “cells”, each of which is a regular octahedron; the Weyl group of is its group of symmetries, just like the Weyl group of was the group of symmetries of the hexagon.
Also like the case, the root system is isomorphic to its own dual. The long roots stay the same length when dualized, while the short roots double in length and become the long roots of the dual root system. Again, a scaling and rotation sends the dual system back to the one we constructed.
The root system is, as we can see by looking at it, closely related to the root system. And so we start again with the -dimensional subspace of consisting of vectors with coefficients summing to zero, and we use the same lattice . But now we let be the vectors of squared-length or : or . Explicitly, we have the six vectors from — , , and — and six new vectors — , , and .
We can pick a base . These vectors are clearly independent. We can easily write each of the above vectors with a positive sign as a positive sum of the two vectors in . For example, in accordance with an earlier lemma, we can write
where after adding each term we have one of the positive roots. In fact, this path hits all but one of the six positive roots on its way to the unique maximal root.
It’s straightforward to calculate the Cartan integers for .
which shows that we do indeed get the Dynkin diagram .
And, of course, we must consider the reflections with respect to both vectors in . Unfortunately, computations like those we’ve used before get complicated. However, we can just go back to the picture that we drew before (and that I linked to at the top of this post). It’s a nice, clean, two-dimensional picture, and it’s clear that these reflections send back to itself, which establishes that is really a root system.
We can also figure out the Weyl group geometrically from this picture. Draw line segments connecting the tips of either the long or the short roots, and we find a regular hexagon. Then the reflections with respect to the roots generate the symmetry group of this shape. The twelve roots are the twelve axes of symmetry of the polygon, and we can get rotations by first reflecting across one root and then across another. For example, rotating by a sixth of a turn can be effected by reflecting with the basic short root, followed by reflecting with the basic long root.
Finally, we can see that this root system is isomorphic to its own dual. Indeed, if is a short root, then the dual root is itself:
On the other hand, if is a long root, then we find
and so the squared-length of is . These are now the short roots of the dual system. Scaling the dual system up by a factor of and rotating of a turn, we recover the original root system.
Before we continue constructing root systems, we want to stop and observe a couple things about transformations of Dynkin diagrams.
First off, I want to be clear about what kinds of transformations I mean. Given Dynkin diagrams and , I want to consider a mapping that sends every vertex of to a vertex of . Further, if and are vertices of joined by edges, then and should be joined by edges in as well, and the orientation of double and triple edges should be the same.
But remember that and , as vertices, really stand for vectors in some base of a root system, and the number of edges connecting them encodes their Cartan integers. If we slightly abuse notation and write and for these bases, then the mapping defines images of the vectors in , which is a basis of a vector space. Thus extends uniquely to a linear transformation from the vector space spanned by to that spanned by . And our assumption about the number of edges joining two vertices means that preserves the Cartan integers of the base .
Now, just like we saw when we showed that the Cartan matrix determines the root system up to isomorphism, we can extend to a map from the root system generated by to the root system generated by . That is, a transformation of Dynkin diagrams gives rise to a morphism of root systems.
Unfortunately, the converse doesn’t necessarily hold. Look back at our two-dimensional examples; specifically, consider the and root systems. Even though we haven’t really constructed the latter yet, we can still use what we see. There are linear maps taking the six roots in to either the six long roots or the six short roots in . These maps are all morphisms of root systems, but none of them can be given by transformations of Dynkin diagrams. Indeed, the image of any base for would contain either two long roots in or two short roots, but any base of would need to contain both a long and a short root.
However, not all is lost. If we have an isomorphism of root systems, then it must send a base to a base, and thus it can be seen as a transformation of the Dynkin diagrams. Indeed, an isomorphism of root systems gives rise to an isomorphism of Dynkin diagrams.
The other observation we want to make is that duality of root systems is easily expressed in terms of Dynkin diagrams: just reverse all the oriented edges! Indeed, we’ve already seen this in the case of and root systems. When we get to constructing and , we will see that they are self-dual, in keeping with the fact that reversing the directed edge in each case doesn’t really change the diagram.
As we did for the series, we start out with an dimensional space with the lattice of integer-coefficient vectors. This time, though, we let be the collection of vectors of squared-length or : either or . Explicitly, this is the collection of vectors for (signs chosen independently) from the root system, plus all the vectors .
Similarly to the series, and exactly as in the series, we define for . This time, though, to get vectors whose coefficients don’t sum to zero we can just define , which is independent of the other vectors. Since it has vectors, the independent set is a basis for our vector space.
As in the and cases, any vector with can be written
This time, any of the can be written
Thus any vector can be written as the sum of two of these vectors. And so is a base for .
We calculate the Cartan integers. For and less than , we again have the same calculation as in the case, which gives a simple chain of length vertices. But when we involve things are a little different.
If , then both of these are zero. On the other hand, if , then the first is and the second is . Thus we get a double edge from to , and is the longer root. And so we obtain the Dynkin diagram.
Considering the reflections with respect to the , we find that swaps the coefficients of and for . But what about ? We calculate
which flips the sign of the last coefficient of . As we did in the case, we can use this to flip the signs of whichever coefficients we want. Since these transformations send the lattice back into itself, they send to itself and we do have a root system.
Finally, since we don’t have any restrictions on how many signs we can flip, the Weyl group for is exactly the wreath product .
So, what about ? This is just the dual root system to ! The roots of squared-length are left unchanged, but the roots of squared-length are doubled. The Weyl group is the same — — but now the short root in the base is the long root, and so we flip the direction of the double arrow in the Dynkin diagram, giving the diagram.