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This is about Deligne's "La série exceptionnelle de groupes de Lie, C.R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 4, 321–326".

When this came out, that was quite something! People were often talking about it, at lunch or coffee time - even making bets if I remember well.

I totally lost contact with all this, since then, now I see that there is a lot of literature on all this, for the most very technical, and I was curious: is there any survey paper or blog post or anything, which explains (for non-specialists) the status of that conjecture of Deligne? I mean, if that mysterious category exists, and what is it.

Many thanks.

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I think that Dylan Thurston disproved something related to that conjecture... but I don't know the details. – André Henriques Jan 10 '13 at 14:45
Can I advertise my paper which explains why one series does not exist? It is one of my unfulfilled ambitions to repeat this exercise for the other series. – Bruce Westbury Jan 10 '13 at 16:59
@Z254R I moved on from this topic because I did not get much response, although I still find it fascinating. You should accept an answer; for one thing, if you don't this will be on record as not having got a satisfactory answer. – Bruce Westbury Jan 10 '13 at 21:35
up vote 5 down vote accepted

My understanding is that the series corresponding to the first three rows of the magic square do not exist but the existence of the series for the last row, the original exceptional series, is still open. This follows from Dylan Thurston's (unpublished) computer calculation and from Pierre Vogel's (unpublished) papers.

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Isn't the subexceptional (E7) series also still open? – Noah Snyder Jan 10 '13 at 16:32
@Noah It is correct that Dylan did his calculation for the first two rows. However you can see a phenomenon in the first three rows but not the fourth which is, I believe, responsible. I also believe that Vogel's work rules out an E7 series. – Bruce Westbury Jan 10 '13 at 16:55
@Bruce, can you give more details on the E7 case? I thought that was still open. – Dylan Thurston Jul 28 '13 at 13:23

The key papers to read about this are Vogel's who conjectures an even better 2-dimensional family which includes all simple Lie algebras, not just the exceptional ones. Deligne's family would correspond to a certain line in this plane. Personally, I find Vogel's papers difficult to read, but you have to read them if you really want to understand this stuff. Some good expositional sources are Cvitanovic's Bird Tracks and Introduction to Vassiliev knot invariants by Chmutov-Duzhin-Mostovoy. In particular, Cvitanovic developed some of the key ideas independently.

The basic idea is that you write down diagrammatically the universal metric Lie algebra object in a symmetric tensor category, using the category of Jacobi diagrams. This is very closely related to Vassiliev's finite type invariants by work of Bar-Natan. The resulting Lie algebra object won't be simple, but you can add the assumption that it is simple by assuming that certain Hom spaces are 1-dimensional. Roughly, conjecturally this added assumption gets you down to a 2-dimensional space of possible simple metric Lie algebra objects.

To get from the universal Lie algebra in the last paragraph to Deligne's exceptional family, you need to impose one more relation which is satisfied by the exceptional Lie algebras. Namely there is a certain invariant vector in the 4th power of the adjoint representation of any exceptional Lie algebra, and this gives an extra relation. This relation is written down explicitly on the first page of Dylan's paper mentioned in comments.

So where are we stuck? Well there's two problems:

  • Do the relations that we can write down suffice to calculate the value of all closed diagrams?
  • Are the relations consistent for all values of the parameters? That is, if you evaluate the same closed diagram in two different ways do you get the same answer?

The first of these questions should be easier than the second. But both are wide open.

What Dylan did in the paper mentioned in comments was look at some analogous situations and calculated directly that it appeared the answers to the first question in those circumstances was yes, but the answer to the second question in those situations was no, and in fact the relations imposed a polynomial relation on the parameter forcing you back to finitely many possibilities. But for Deligne's question, the calculation would require looking at very large examples which were beyond the computer's power.

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Another source I found helpful was the papers of Kneissler though they are somewhat technical. – Noah Snyder Jan 10 '13 at 16:39

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