Background reading: John Stembridge's webpage.

The idea is that when you want to prove a theorem for all root systems, sometimes it suffices to prove the result for the simply laced case, and then use the concept of folding by a diagram automorphism to deduce the general case.

I have never seen an example of this in practice. So my question is: What are some (good) examples illustrating this technique?

up vote 12 down vote accepted

One very important use of this technique is the relation between Lie algebras / quantum groups and quiver varieties. I first saw something about this in Lusztig's book, Introduction to Quantum Groups; but see also this arXiv paper Alistair Savage. Quiver varieties are important for categorifying many structures related to a simple Lie algebra (its representation theory, its enveloping algebra, etc.). Categorification is a long story that leads to all kinds of interesting things, and it is a sequel to the long story of quantum groups themselves. But even if you're not learning about either one for their own sake, Lusztig already needed it to prove properties of his canonical bases of representations of simple Lie algebras.

A Dynkin-type quiver is an orientation of a Dynkin diagram. A quiver representation is a collection of maps between vector spaces in the pattern of the diagram. A quiver variety is then a variety of (certain of) these representations, for fixed choices of the vector spaces. The point is that you can only define a quiver for a simply laced Dynkin diagram. You need the folding automorphism to obtain quiver varieties or information from quivers in general in the multiply laced case.

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    Although, you can get categories of quiver representations which correspond to any finite crystallographic type if you're prepared to work over a non algebraically closed field. I don't know if this story extends to quiver varieties. – Hugh Thomas Nov 4 '09 at 17:54

A more basic example than Greg's would be the construction of simple Lie algebras associated to a Dynkin diagram: you can first construct the simply-laced ones, and then exhibit the remaining ones as the fixed points of diagram automorphisms. Since the simply-laced ones can be constructed easily from the root lattice, this is one of the cleanest proofs of the existence theorem -- it's discussed in chapter 7 of Kac's book. (Kac also uses diagram automorphisms to construct the twisted affine Lie algebras in the following chapter, which gives another example of the technique.)

  • This certainly counts as a case when you can do it, but not quite as a case when you "need" it. Although arguably any time you do "need" it, it is an invitation to find a more direct argument. – Greg Kuperberg Nov 30 '09 at 15:56
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    The example of affine algebras is perhaps in that case: the loop realization allows you to calculate the roots of the affine Lie algebras, and you need the diagram automorphisms of the finite-type algebras in order to get all the affine types. I don't know of another way to calculate the roots and their multiplicities. – Kevin McGerty Nov 30 '09 at 20:10

Another example of this technique is in chapter 7 of Kostant modules in blocks of category $\mathcal{O}_S$ by Boe and Hunziker. They first classify the said modules in the simply laced case and then use diagram folding (and an associated isomorphism of partial flag manfiolds) to deduce the theorem in the general case.

This is by now an old question, but a more recent example is given in a preprint by a current MIT graduate student here. I think the moral of the story is that there are quite a few directions in Lie theory where folding plays a significant role in getting from the simply-laced case to other cases.

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