How do Jordan algebras help one understand representations of exceptional Lie algebras?

For this question I'm happy to take the complex numbers as the base field.

I've been trying to learn a little bit about the exceptional Lie algebras and for a while they seemed inaccessible. I looked at McCrimmon's nice survey on Jordan algebras: http://projecteuclid.org/euclid.bams/1183540925 and it seems at least that one can use these "new" (in the sense that they are new to me) gadgets to get a foothold on the exceptionals by realizing them as automorphisms, or derivations, or whatever of basically the same guy $H_3({\bf O})$ (Hermitian $3 \times 3$ octonion valued matrices). It seems interesting but since I also have no intuition for these new objects, I'd like to ask the following question:

Are there any available written accounts of using "Jordan theory" to get working models for the representations of the exceptional Lie algebras, or at least their fundamental representations?

This is coming from the perspective of someone who likes explicit constructions and being able to write down bases, matrices, etc. if desired.

I can at least appreciate the case of $G_2$: using the octonions, we get its minimal representation, and the other fundamental is its adjoint, but we can also get all of its irreducibles by applying Schur functors to its minimal and mimicking Weyl's "tracefree tensor" construction (Huang, Zhu, "Weyl's construction and tensor product decomposition for $G_2$ http://www.jstor.org/pss/119028). Okay not quite Jordan algebraish, and I don't expect something like this to be written down for other types, but hopefully this gives an idea of what I'm looking for.

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In your header, maybe it's better to omit the first word "How"? I haven't spent enough time with the literature on explicit constructions of representations (going back to E. Cartan) to be confident about the answer one way or the other beyond what you can do with the adjoint representation and sometimes a smaller fundamental representation. As Skip points out, there are good treatments in the modern literature of ways to construct the Lie algebras themselves and their forms over various fields. But representation theory involves understanding all fundamental representations. –  Jim Humphreys Sep 27 '10 at 18:17
P.S. You may or may not find the scattered literature in mathematical physics journals helpful for concrete constructions of representations. But proceed with those papers at your own risk and check the methodology. –  Jim Humphreys Sep 27 '10 at 18:19

There are descriptions for all the exceptional Lie algebras in terms of some sort of nonassociative algebra, except for $E_8$. (For $E_8$, you can say that it is the derivations of itself as a Lie algebra, bu this is hardly informative.) In each case, one describes the Lie algebra or group as derivations or automorphisms of some algebraic structure on the smallest-dimensional irreducible representation of the algebra, roughly speaking.

• Type $G_2$: automorphisms of the octonions, as you say
• Type $F_4$: automorphisms of an Albert algebra (27-dimensional exceptional Jordan algebra), see e.g. Chapter IX in The Book of Involutions.
• Type $E_6$: easiest is to think of this as the group of norm isometries for an Albert algebra, which is totally sufficient for the complex numbers.

Up to here, the book "Octonions, Jordan aglebras, and Exceptional groups" by Springer and Veldkamp is a good reference.

• Type $E_7$: Stabilizer of a quartic form and skew-symmetric bilinear form (making a "Freudenthal triple system" - this phrase will get you to a list of references) on its 56-dimensional irreducible representation.

My paper Structurable algebras and groups of type E6 and E7 gives a survey on a kind of nonassociative algebra that links types E6 and E7 in the same way that types F4 and E6 are linked (smaller one is an automorphism group, larger one is the isometries of a norm form). You can see this from a different view in Springer's paper Some groups of type E7. Or from the view of representation theory in Helenius's paper Freudenthal triple systems by root system methods.

Now a disclaimer. You asked your question for the complex numbers, but I think this Jordan-theoretic interpretation does not have much value there because you already have the root space decomposition of the Lie algebra, etc. Rather, the purpose of this viewpoint is to handle more general base fields and especially to handle anisotropic (over the real numbers, one usually says "compact") groups over those fields. That's the whole idea behind the Book of Involutions: to use similar interpretations to study classical groups over general fields.

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These are useful comments and references, but don't directly address the broader question asked about representation theory (say over the complex numbers). –  Jim Humphreys Sep 27 '10 at 19:43
I think the Jordan theory viewpoint gives a nice alternative presentation of the smallest fundamental representation; it is a nice counterpoint to the traditional Verma-module view. But I don't get much out of it for understanding the other representations. On a different but related topic, the Jordan theory view can be used to give concrete descriptions of the projective homogeneous varieties, in much the way that you talk about quadrics for SO_n or Grassmannians for SL_n. –  Skip Sep 28 '10 at 15:28
Skip, thanks for the answer. Could you provide some references to how Jordan theory helps one think about the homogeneous varieties? It's presumably more complicated over non algebraically closed fields, so over C is enough for me. –  Steven Sam Sep 28 '10 at 21:12
The situation is totally analogous to the classical groups. I describe the "minimal" homogeneous varieties, which correspond to single vertices in the Dynkin diagram. For SL_n each of them is a k-Grassmannian, meaning a variety whose points are the k-dimensional subspaces of C^n. (The others are flags of these things.) Similar descriptions work for types B, C, D. For G2, there are two minimal varieties, whose C-points are the 1-dimensional and 2-dimensional subspaces of the (complex) octonions on which the multiplication is identically zero (compare Problem 23.54 in Fulton & Harris). –  Skip Sep 29 '10 at 13:53
(continued): You can find scattered across the literature descriptions of the varieties for F4, E6, E7. Since you are interested in C, probably everything is in Freudenthal's papers "Beziehungen...". In references, typically it is easy to identify what the description of the flag variety should be, but the proof of the description is missing. You can find such proofs for E6 and E7 in my "Structurable algebras" paper, but a better, root-system-theoretic presentation is also in Carr-Garibaldi "Geometries..." dx.doi.org/10.1016/j.exmath.2005.11.001 –  Skip Sep 29 '10 at 14:01

No, you cannot use "Jordan theory" to get working models for the representations of the exceptional Lie algebras. There is a chance of doing it with $F_4$ but not with any $E_n$-s.

Historically, people used Jordan algebras to prove existence of exceptional Lie algebras but there are better ways nowadays such as Serre's relations or Freudental's magic squares...

Let me add that Jordan algebra is just a (2,1)-tensor preserved by the group. A more natural question is what tensors the group is a stabilizer of.

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