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Edit:

Having no luck with my previous formulation of my question (too focused on the E6 case I think), I have decided to reformulate it the following way:

1/ Is there an easy way to build the Satake diagrams from any Dynkin one? (I mean without too much calculations ...)

2/ Especially, is there an easy way to tell which Satake diagram type of construction are not allowed depending on the lie algebra considered?

3/ From a Satake diagram, considering it determines the real form of the Lie algebra uniquely, how can we recover it? For instance how can we get the multiplicity of the restricted roots?

Do you know any paper/book where this is clearly described?

Old formulation:

I am learning how to classify non complex real semisimple Lie algebras. I understand what the Satake diagrams are supposed to describe, but I am having problems when I try to build them on a simple case like for the exceptional algebra E6.

Apart from the compact form (all black vertexes) there are only four forms described by certain Satake diagrams. If I try to build them “naively” by hand from the Dynkin diagram I found more than four.

Can somebody explain to me how I get rid of the wrong ones (in that special case E6 for instance)? Is there any way to do it without too much calculations from the Dynkin diagram of E6? Or just give me the appropriate pdf to look at? I could not find any!

brunoh
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  • For real forms of $E6$ see this paper: "Satake diagrams of the real forms $e_{6,−26}, e_{6,−14}, e_{6,2}$ are carefully developed". – Dietrich Burde Aug 24 '18 at 18:07
  • @DietrichBurde thank you for this reference paper. Unfortunately I already knew it and read it. It is about building actual realizations of the three real forms of E6, and then checking that their Satake diagram is conformed. Nothing in it about how to build the Satake diagrams by themselves (meaning how to get rid of all the "naively" possible ones). My question is an order of magnitude simpler that the content of this paper :=) – brunoh Aug 24 '18 at 18:47

1 Answers1

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Upshot: You should not expect too easy answers to these questions.

I assume you only work over the real numbers $\Bbb R$, and you turn a Dynkin diagram into a Satake diagram by colouring some vertices black and adding some arrows between vertices. If such a diagram reprsents an actually existing real form, I call it "admissible". Now there are two relatively easy facts which exclude many of these "naively possible" Satake diagrams from being admissible:

A. In the cases $A_n$, $D_n$ for odd $n$ (!) and $E_6$, the diagram has to be invariant under the obvious reflection.

B. If you erase any white vertex and everything that is connected to it by an arrow, what remains must be an admissible Satake diagram again.

Note as a third fact, which conversely shows existence of certain forms, that

C. Appropriately "patching" admissible Satake diagrams gives admissible diagrams.

But already the rigorous technical statements, and proofs, of these three facts need quite some machinery from the structure theory.

Further, although A-C alone can reduce the number of naively possible diagrams by a great deal to non-naively possible diagrams, these few remaining possible cases are very intricate to check. This can be seen from the fact that one can do the same theory over number fields or $p$-adic fields instead of $\Bbb R$, and some Satake diagrams which are admissible over $\Bbb Q_p$ or $\Bbb Q$ are not admissible over $\Bbb R$, or vice versa. Which shows that in the end, one has to use some arithmetic properties of the ground field instead of the more general techniques on which A-C rely. The $E_6$ case e.g. is quite nasty, as the (existence of the) paper quoted by D. Burde shows.

Blowing my own horn, I went through those discussions in length (but focussing on the $p$-adic case) in my doctoral thesis:

Semisimple Lie Algebras and their classification over p-adic Fields, Mémoires de la SMF 151 (2017).

For the real case which seems to be your interest, there is an old paper which went through this almost entirely with combinatorics of the diagrams,

Shoro Araki: On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ., Volume 13, Number 1 (1962), 1-34.

A mixed approach is taken in Satake's book (in particular the appendix by M. Sugiura),

Ichiro Satake: Classification theory of semi-simple algebraic groups, M. Dekker, New York 1971

Added: Compare also (with a focus on the $E_6$ case, but touching on the general ideas) question and answers in https://mathoverflow.net/q/298948/27465.

Torsten Schoeneberg
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  • My comments on your answer are: 1/ you answered exactly what I was asking & more 2/ perfectly crafted & written 3/ add very interesting points (for instance your A/B/C criteria & your comment on the importance of the ground field own idiosyncrasies ...) 4/ your choice of references is plenty sufficient and to the point. So in a word: perfect answer. Just one little question remaining: is this D criteria also valid: erasing arrows and black vertices must give an existing Dynkin diagram? Thank you for your time! – brunoh Aug 26 '18 at 13:11
  • BTW, I just ordered your smf memoir (good savings because I am member of smf). Looks so interesting! :=) – brunoh Aug 26 '18 at 13:29
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    To your proposed "D" criterion: No, this is not valid. Example: Take the derived Lie algebra of $\mathfrak{gl}_2(\Bbb H)$ where $\Bbb H$ are the Hamilton quaternions. That is a form of type $A_3$ where the Satake diagram has the two outer vertices black, and the inner vertex white. Erasing just the left (or just the right) black vertex gives a Satake diagram of type $A_2$ with one black and one white vertex, which is forbidden (over any field) by criterion A. – Torsten Schoeneberg Aug 26 '18 at 21:25
  • yes you are right ... but my proposed D criterion was to get rid of all arrows & black vertices at once. It should work because I understood that on the non compact algebra the restricted roots are a root system right? – brunoh Aug 26 '18 at 21:29
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    The restricted roots form a (possibly non-reduced i.e. containing summands of type $BC_n$) root system, but you do not just get a diagram of that root system by erasing all black vertices and arrows from the Satake diagram (e.g. then one could never get such a $BC_n$ system); the computation (which I think goes back to Tits) is a little more intricate, but can usually be done quite easily once one knows how, and is covered in my thesis. -- [... to be continued] – Torsten Schoeneberg Aug 26 '18 at 21:48
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    [continued] That being said, I do not see a counterexample to such a criterion D you propose, but it seems to me as if that is just a special case of: If you erase any set of vertices (and edges touching them) from a Dynkin diagram, you get a (disjoint union of) Dynkin diagram(s) again, which IIRC is a well-known property of Dynkin diagrams used in the standard classification proofs of those. – Torsten Schoeneberg Aug 26 '18 at 21:50
  • thanks again for your time. I will mull over your last comment. – brunoh Aug 26 '18 at 22:13