There are many ways I could proceed from here, all of which feel like a radical shift. But my goal was Artin-Wedderburn with some applications to representations and group rings, so probably the most important concept of this sequence of posts hasn’t been mentioned at all. This is the notion of being semisimple.

We’ll work from the definition that an R-module, M, is semisimple if every submodule is a direct summand. There are many equivalent ways of thinking of this.

First, note that a submodule of a semisimple module is semisimple. This just requires justifying that intersecting works nicely, and it does (a pretty straightforward exercise if you want to try it). An often useful equivalent condition for semisimple is that M is a direct sum of simple submodules.

The definitions I really want to get to are about rings (which is why I sort of breezed through that first part). A ring R is semisimple if it is a semisimple module over itself. But note that the submodules of R are just the left ideals, so R is semisimple iff every left ideal is a direct summand.

In fact, we have the following equivalent statements:
1) Every R-module is semisimple.
2) R is a semisimple ring.
3) R is a direct sum of a finite number of minimal left ideals.

Proof: is trivial. For , we know that the simple submodules of R are the minimal left ideals of R, so where is minimal. So we just need this sum to be finite. But we know that , a finite sum where (reindex if you want rigor with indices since isn’t necessarily a subset of ). So for any , we have . i.e. . So .

Now we said that a direct sum of minimal left ideals (simple submodules) was an equivalent definition of semisimple, so . So for , let M be an R-module with R semisimple. Since any R-module is an epimorphic image of a free module, say . But each , so they are semisimple. Thus F is semisimple. But then M is a semisimple module.

With a view towards Artin-Wedderburn, I’ll present what is probably the most important example of a semisimple ring.

Let be a division ring. Then the claim is that is semisimple. Let . i.e. we have as the matrix with zeros everywhere except the i-th column, which can be anything from the division ring. Certainly, . And also each is a left ideal. But why are they minimal?

Suppose some left ideal is properly contained in . Then there is some . So take matrix , we can easily form a matrix such that (since is a division algebra) which contradicts L being a left ideal. Thus the are minimal and hence is a semisimple ring.

Well, things can get ultra busy around mid-terms. I don’t think I’ve posted in two weeks. What I really wanted to do next was to post some category theory basics. I’m not sure if I should, though, since so many math blogs have already done this. I then wanted to go on to define the fundamental group purely in categorical language. It turns out to be a really nice construction compared to the tedious typical way.

Instead, I’ve recently become quite interested in rings. There also seems to be a very large lack of “pure” ring theory in the blogosphere. Sure rings pop up and are needed by people doing things with algebraic geometry, say, but using ring theory isn’t the same as developing it.

I’m going to cover the basics quite quickly with the assumption of previous exposure, since I really want to get to some of the more interesting constructions (i.e. localization), then I’ll slow it down.

Ring: We have a set with two operations, we’ll call them addition and multiplication. The addition part forms an abelian group, and the multiplication…well, it puts you back in the set and is associative. We need a way to relate these operations, so we also require that and , i.e. there is a distributive law in effect. Note that there is not required in general to be a multiplicative identity, multiplicative inverses, or commuting of the multiplication.

NOTE: Until I say otherwise I will assume the ring is commutative (meaning multiplication) with 1 (meaning having a mult identity). will always denote this.

Subring: A subset of that is itself a ring.

Ideal: A subring that “swallows” multiplication. So is an ideal if for any we have that for all .

Prime ideal: An ideal is prime if for element we have that either or .

Principal ideal: An ideal is principal if it is generated by a single element. So an ideal is generated by “a” if .

Maximal ideal: A proper ideal such that there is no other ideal with the property (where all containments are proper).

Domain: A ring in which the cancellation law holds. i.e. if and , then . Note that no element can divide zero, so if , then either or .

We can quotient in the natural way: is the set of cosets of I where our operations are and . We get the nice result that any ideal of is of the form where is an ideal of (and ).

I think that may lay down all the terminology I’ll need to get started. I’m not sure if I’ll really use any of these terms for awhile, though.

Common rings: . Note that we can get from by taking “quotients.” This can be made precise for any domain. It is called the fraction field of denoted .

Let me take some time to explain this, since it is the motivation for localization. We want to form a field that contains as a subring such that the elements of , say have the form where . Note that this “looks” like division, and in fact is division in the case of .

To make this process precise takes a bit of work, though. Set up . Define (a,b)~ (c,d) iff . This is done since we want our relation to look like fractions, a/b , so a/b=c/d if we can cross-multiply and get the same thing. It is straightforward to check that this defines an equivalence relation. Now we let be the set of equivalence classes.

Our operations on should mimic those of fractions, so our addition is [a,b]+[c,d]=[ad+bc, bd] and [a,b][c,d]=[ac,bd]. These are well-defined and it is just computation to check that the axioms of a field are satisfied. (If you want a hint: the zero is [0,1] and the 1 is [1,1], the additive inverse of [a,b] is [-a,b] and the mult inverse is [b,a]).

Before finishing up, I want to point out how restrictive we had to be. We want a more general way of doing this. We don’t want to require that R be a domain, and we don’t want to have to take fractions with every single element in R. It turns out this general process is extremely useful and it is called localization.

Let’s actually try to make some progress on QFT today. There are three parts to make a minimal definition. First, you need a module D over a *-commutative ring. So to get a few definitions on the table. A *-ring, R, is pretty easy. You just have a ring with an antiautomorphism and involutive mapping . This means that (i) , (ii) , (iii) , and (iv) . So if you’ve seen rings, this shouldn’t be out of grasp. An example would be complex numbers with complex conjugation. A <a href=”http://en.wikipedia.org/wiki/Module_(mathematics)”>module</a> is basically a generalization of a vector space.

The second part is a Hermitian inner product . So recall that Hermitian just means that it is self-adjoint. You could think of this as when you express the operator as a matrix the conjugate transpose is itself again. Lots of operators satisfy this, like the differential operator. Essentially the property Hermitian is in place, because if something is obsevable then it is Hermitian.

The last part is that we need a *-algebra, , of operators acting on D. Let’s jump out to a bigger picture for a second. The details here are sort of the details of getting around a problem. What we really want is basic. We want a Hilbert space H and an operator satisfying the axioms we want. So our field , and our operator defined at each as (an operator on H). The problem we are skirting is one of how to get around when x and y get arbitrarily close (an uncertainty problem as you might guess).

So we do the standard trick of “smoothing out the singularities.” Instead of points we will use bump functions. A bump function on M is just a smooth function with compact support. We redefine the operator then to be . Here is why I jumped out to the big picture we are skirting around. is generated by .

Some examples will be instructive. Let G be a group and D an orthogonal representation. Then is the group-ring of G, with “*” as . Or we could let L be a Lie algebra acting on a vector space D with an invariant symmetric inner product. The algebra can be with . Or we could take as any -algebra or von Neumann algebra and D any Hilbert space that is a *-representation.

These three examples should make us notice something. These are not things physicists typically work with (unless they are doing mathematical foundations of QFT or something). So despite having a definition in place, we might need to make some restrictions or correlations to what computations are being made down the road. These three examples are QFT’s, but that is sort of weird, since we usually speak of “QFT” and not “a QFT” or “this QFT” as if there is only one.