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I apologize for this being a very very vague question.

Just as personal experience, I never feel that I fully grasped the theory of root systems in Lie algebras and Lie/algebraic groups (I shall call these $\textbf {LAG} $objects for short). My "psychological" response to these objects seem to be--they are really "ugly" and unnatural, where do they come from?? Of course, we can say, OK, let's work out the theory of $sl_2$, then everything is just "natural" generalization of easier cases. But this explanation just doesn't satisfy me.

I've been worrying about such things for really quite a long time. And probably the real question is, do these root systems have some (really nice, really simple) geometrical interpretation? For example, is it possible that they are somehow related to some nice topological space, some sheaf, some cohomology, etc...? And do they have some nice counterparts in other branches of mathematics? (it seems to me they only appear in places like LAG objects), do they show up in some other (probably surprising) places?

Actually, as we know, study of LAG objects themselves are closely related to their representation theories. And their representation theory are also obejects that seem hard to visualize in some "easy" geometrical way. I know we can probably put these things as a category and do some abstract algebraic geomtry with them. But still, we cannot avoid using these ad-hoc techniques with them. (like root systems, Dynkin diagrams, etc, etc)

Perhaps I should stop here. In any case, I hope this question will not be closed as a spam. And hope some one can understand some of my confusion and shed some light (geometry) on it. Thank you.

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Dynkin diagrams and their friends show up all over the place in weird spooky ways. Go look up the [ADE classification]. –  Charles Rezk Oct 20 '10 at 23:49
    
I have found Peter Woit's exposition of this material in these notes to be quite good: math.columbia.edu/%7Ewoit/repthy.html –  Kevin H. Lin Oct 21 '10 at 0:00
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(For future reference, you can get boldface by using double asterisks or double underscores: __LAG__ gives LAG and **LAG** gives LAG. No need to use TeX, which requires the reader's browser be configured correctly (it usually is) --- Markdown runs on the server, and hence is faster.) –  Theo Johnson-Freyd Oct 21 '10 at 2:03
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Wasn't it Von Neumann who said that in mathematics you don't understand things, you just get used to them? –  Angelo Oct 21 '10 at 2:59
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I suggest viewing Kubrick's movie "Dr. Strangelove: How I learned to stop worrying and love the bomb". –  Lee Mosher Nov 8 '13 at 13:17

6 Answers 6

One thing to keep in mind is that the process that starts with LAGs and ends up with root systems is not "functorial". To keep everything very definite, let me restrict attention to finite-dimensional semisimple Lie algebras over $\mathbb C$. Then there are maps {isomorphism classes of s.s. complex Lie algebras}{isomorphism classes of Dynkin diagrams}, and at the level of isomorphism classes, the two maps are inverse to each other. In fact, there is a wonderfully functorial map going ←, i.e. from Dynkin diagrams to algebras, which was worked out by Serre, I think (maybe Chevalley). But the → map requires making all sorts of choices: pick a Cartan subalgebra, pick a notion of "positive" for it. Let $\mathfrak g$ be the Lie algebra and $\mathfrak h$ the chosen Cartan; then the group $\operatorname{Aut}(\mathfrak g)$ acts transitively on choices of simple system, and the stabilizer is precisely $\mathfrak h$. (Or, rather, in $\operatorname{Aut}(\mathfrak g)$ there are "inner" automorphism and "outer" automorphisms, and the inner ones in fact act transitively, and $\operatorname{Out}/\operatorname{Inn}$ acts as non-trivial diagram automorphisms. By "precisely $\mathfrak h$" I mean that the stabilizer is $\exp\, \mathfrak h \subset \operatorname{Inn}$.) So the space of choices is a homogeneous space for $\operatorname{Inn}(\mathfrak g)$ (which is the smallest group integrating $\mathfrak g$) modeled on $\operatorname{Inn}(\mathfrak g)/\exp(\mathfrak h)$. But, anyway, the point is that $\exp(\mathfrak h)$ acts nontrivially on $\mathfrak g$ but, as I've said, trivially on the Dynkin diagram, and hence trivially on the group that you construct from the Dynkin diagram.

This might be why you don't like the notion of root systems: you really do need to make choices to identify algebras with their root systems. It's something like picking a basis for a vector space — great for computations, but not very geometric. As a precise example: for $\mathfrak{sl}(V) = \{x\in \operatorname{End}(V) \text{ s.t. }\operatorname{tr}(x)=0\}$, a choice of root system is the same as a choice of (ordered) basis for the vector space $V$.

On the other hand, I claim that you should like the representation theory of a LAG. One way to study this representation theory (I might go so far as to say "the best way") is to pick a root system for your LAG and look at how $\mathfrak h$ acts, etc. Then, for example, finite-dimensional irreducible representations of a semisimple Lie algebra $\mathfrak g$ correspond bijectively with ways to label the Dynkin diagram with nonnegative integers. So you can really get your hands on the representation theory.

But representations of a group $G$ is a very geometric thing. There is some sort of "space", called "$BG$" or "$\{\text{pt}\}/G$", for which the representations of $G$ are the same as vector bundles on this space. If you don't like thinking about "one $G$th of a point", there are homotopy-theoretic models of $BG$.

You should also think of categories as geometric. Think about the case when $G$ is a (finite) abelian group. Then you might remember Pontrjagin duality: the irreducible representations of $G$ are the same as points in the dual group $G^\vee$. Then, at least for $G$ a semisimple LAG, you might think of its finite-dimensional representations as being like the points of some "space" $G^\vee$. The difference is that in the abelian case, all the points on $G^\vee$ correspond to one-dimensional representations, whereas in the semisimple nonabelian case in general the points are "bigger". The points are parameterized by the positive weight lattice, but they aren't actually the positive weight lattice. But the space "$G^\vee$" is some space noncanonically-isomorphic to the positive weight lattice. Again, this is like how the Euclidean plane is noncanonically isomorphic to the Cartesian plane.

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The isomorphism with the positive weight lattice is actually canonical. There is a notion of the abstract Cartan (eg in Chriss and Ginzburg), the point being that for any two choices of Borel B, if N is its unipotent radical, the two quotients B/N are canonically isomorphic. So while it may look as though you're choosing a Borel to define the postive weight lattice, looking at things from this `abstract' Cartan pov means the constructions are canonical. –  Peter McNamara Oct 22 '10 at 18:53

There is a limit to how simple and natural you can make root systems look, given that there exist the sporadic examples. It's hard to imagine arriving at the full classification without rolling up your sleeves to do some explicit computations at some point.

I think the better way to approach the subject is to begin not with the root systems themselves but some other natural question, such as "What are all the finite-dimensional simple Lie algebras over $\mathbb{C}$?" Trying to answer this question, you find yourself forced to invent root systems.

Will it help you psychologically to consider the analogous problem of classifying all finite simple groups? One might similarly say that all those sporadic groups are ugly and unnatural—where do they come from?? But once you set yourself the task of classifying something, you're committed to including everything that shows up, no matter how unusual it looks. As for ugliness, beauty is in the eye of the beholder. Sporadic groups (and their root systems) have a tendency to look more and more beautiful the more you study them.

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@Timothy. Thank you very much for your comment. Indeed, when we are trying to work on some question by our own hands, everything that is helpful are in some sense beautiful, since mathematics is such a compatible body of machine. Probably what I really should do is get my hands really really dirty on root systems and related theories, until one day I find them naturally beautiful. :p –  root Oct 21 '10 at 0:39

I had a problem which may be similar when I encountered root systems for the first time. In retrospect, I think that the problem was that I was reading about specific realizations of root systems (eg, the $A_n$ roots embedded into $\mathbb R^{n+1}$ as $e_i-e_j$ , etc.). Those specific realizations looked very ad hoc to me. The point, I would say, is that, while it's good to spend some time with some of those specific realizations, you should get used to thinking about root systems in a uniform way, at which point their power becomes pretty evident. (But maybe this is really a different problem from yours.)

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I don't really understand what you are asking for. Root systems are geometric objects associated to finite reflection groups. Try reading Jim Humphrey's "Reflection Groups and Coxeter Groups" for a very nice and elementary treatment. You can also read the book by Fulton/Harris which deals directly with Lie groups (which, by the way, are about the nicest topological spaces I can think of). If, after reading both of these books you are still unsatisfied, you can move on to the book by Chriss/Ginzburg.

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For a very non-Lie Theoretic reason to care about the ADE root systems, look at http://front.math.ucdavis.edu/0809.2579 and do exercise 66 on DuVal singularities of surfaces. It's good to do by hand if you need practice with blowups, otherwise, just looking at the fact that you get ADE out of canonical surface singularities should motivate the need for Dynkin diagrams, which describe the root systems.

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You might like this, some of Gosset's root lattices showed up as examples over $\mathbf Z$ to Must a ring which admits a Euclidean quadratic form be Euclidean?
especially $E_8,$ as in

http://web.archive.org/web/20130603143515/http://www2.research.att.com/~njas/lattices/E8.b.html

What I did was take the quadratic form given by the included Gram matrix, that form being "improperly primitive." So then I took half of that, which is a quadratic form in eight variables with integer coefficients. I had separate lists for five or fewer variables, I was looking for anything in six or more and came upon these. The trick gave one of Pete's Euclidean quadratic forms for $A_6, \; D_6, \; E_6, \; D_7, \; E_7, \; E_8.$

Here is the index for the lattice pages:

http://web.archive.org/web/20130603053022/http://www2.research.att.com/~njas/lattices/index.html

In case it should be helpful, the fact that $E_8$ worked seems to follow from arithmetic of the octonions. Here we go, On Quaternions and Octonions by John Horton Conway and Derek Alan Smith, Theorem 2 on page 109.

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