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I'm going to say beforehand that this question has something of a "am I missing something?" flavor. I'm in that odd position mathematicians often find themselves, where a topic has been addressed very glancingly in the literature for years, but seemingly without anyone actually dealing with it carefully. One often feels that it can't be that such an basic thing has been ignored so long.

People in Lie theory have known for many decades that a basic and important concept in understanding Dynkin diagrams is "folding." My most basic question is thus:

Has anyone written a careful and systematic treatment of folding for arbitrary (valued) graphs, including the associated embeddings of Lie algebras?

I ask in large part because I need a fair amount of this theory for a paper I'm working on, and don't want to do too much wheel-reinventing or ignoring references that deserve mention. An incomplete list of what I have found:

  1. The closest thing I have found is this survey of Lemay, but it doesn't discuss the embeddings of algebras, since he works on the wrong side of the duality discussed by Humphreys in this question.
  2. I also found this paper of Berenstein and Greenstein useful, but it (and Kac's book, which it references) only deal with the finite type case.
  3. This paper of Naito has the ultimate result I'm looking for in terms of embeddings, but has no details (it's 4 pages long and contains one proof sketch) and no discussion of graphs.
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  • $\begingroup$ If the answer is indeed no, then it is hard to actually give this as an answer, since one can never be 100% sure that the answer is no. Whenever I've written about folding in my own papers, I haven't worried about citing references on folding. Perhaps this makes me a naughty mathematician. $\endgroup$
    – Peter McNamara
    Feb 9, 2016 at 5:00
  • $\begingroup$ @PeterMcNamara Well, it's possible that someone who's deep in the field will know. There are some very specific topics where I'm very confident I'm aware of all the important papers. Anyway, it was definitely naughty if you weren't careful about this Langlands duality issue mentioned in the linked question; I certainly was mixed up about it until recently. $\endgroup$
    – Ben Webster ♦
    Feb 9, 2016 at 5:16

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