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I know that $G_2$ can be described as the subgroup of $SO(7)$ preserving a specific element of $\Lambda^3(\mathbb{R}^7)^*$. It can thus be realized as a matrix group. Prof. Robert Bryant did describe in his answer to the post A question on complex semisimple Lie groups and $(\mathbb{C}^2, \omega)$ a nice description of the complex $E_6$ as the group of symmetries of $V = \Lambda^2_0 (\mathbb{C}^8)^*$ endowed with a cubic form on that space. In that description, what is the real structure which gives the compact real form $E_6$? I conjecture it would be induced by the real structure $j \wedge j$ on $V$, where $j$ is a quaternionic structure on $\mathbb{C}^8$ such that $\omega$ (see Prof. @RobertBryant's answer in the above link) is real. It is just a guess. Is it correct please?

I do not really know how to realize the compact real forms of $F_4$, $E_7$ and $E_8$ as matrix groups. Your help is kindly appreciated. References are more than welcome (especially if they can be found online, and I hope "my" library has access to them!). If someone feels like writing a whole answer, then that would be great too. It is time for me to learn more about the exceptional Lie groups.

One last thing. I know that what I am looking for can be found in the relevant E. Cartan's papers. However, while I would definitely learn a lot by going back to the source, yet I don't have as much free time nowadays as I would like to (not to mention that reading Cartan is known to be difficult, and it is not the language barrier, in my case). So is there a simplified and modernized version of that part of Cartan's work please, that would also discuss compact real forms?

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  • $\begingroup$ Do you mean the simply connected group? the adjoint group? all of them? this strongly affects the question. $\endgroup$
    – YCor
    Commented Jun 22, 2020 at 9:12
  • $\begingroup$ @YCor, I understand. I do not require the compact real form to be simply connected. I just want some compact real form for each type. Actually, Prof. Robert Bryant's answer is what I really want to know/learn. I will try to read Cartan. $\endgroup$
    – Malkoun
    Commented Jun 22, 2020 at 9:42

2 Answers 2

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Cartan describes all of the compact real forms of the simple Lie groups over $\mathbb{C}$ in his first paper that classifies the real forms. In fact, he describes them exactly in the terms that you ask for: A representation of the complex Lie group together with an auxilliary structure, either a real structure on the complex representation space or a Hermitian quadratic form.

For $\mathrm{G}_2$ (resp, $\mathrm{F}_4$, $\mathrm{E}_8$), the compact forms are represented as special orthogonal real matrices of rank 7 (resp., 26, 248). For $\mathrm{E}_6$ (resp. $\mathrm{E}_7$), the compact real forms are represented as special unitary matrices of rank $27$ (resp. $56$).

Explicitly, here are the defining structures in the lowest dimensional representations of the compact real forms of the exceptional groups:

  • $\mathrm{G}_2$ is the stabilizer of a $3$-form on a real vector space of dimension $7$.

  • $\mathrm{F}_4$ is the stabilizer of a quadratic form and a cubic form on a real vector space of dimension $26$. (I believe that the cubic form alone is enough to define $\mathrm{F}_4$.)

  • $\mathrm{E}_6$ is the stabilizer of a cubic form and a positive definite Hermitian form on a complex vector space of dimension $27$. (The cubic form alone only defines the complex $\mathrm{E}_6$.)

  • $\mathrm{E}_7$ is the stabilizer of a symplectic form, a quartic form, and a positive definite Hermitian form on a complex vector space of dimension $56$. (The quartic form and the Hermitian form by themselves are almost enough to define $\mathrm{E}_7$; they define a group with two connected components, the identity component of which is the compact $\mathrm{E}_7$.)

  • $\mathrm{E}_8$ is the stabilizer of a $3$-form on a real vector space of dimension $248$.

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  • $\begingroup$ Ok, I will read Cartan. (French) Après tout, cela fait assez longtemps que je n'ai pas lu un article en français! $\endgroup$
    – Malkoun
    Commented Jun 22, 2020 at 7:21
  • $\begingroup$ Do you think that for the case of $F_4$, some even power of the cubic form may be a power of the quadratic form, or something like that? $\endgroup$
    – Malkoun
    Commented Jun 22, 2020 at 7:29
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    $\begingroup$ Also trying to guess, is the $3$-form nothing but the Cartan $3$-form on the Lie algebra, meaning $(x,y,z) \mapsto \operatorname{tr}(ad_x,[ad_y,ad_z])$? $\endgroup$
    – Malkoun
    Commented Jun 22, 2020 at 8:01
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    $\begingroup$ @QGravity: The best reference I know is Helgason's "Differential Geometry, Lie groups, and Symmetric Spaces". See Table V of Chapter X, Section 6. If all you want is the classification of the compact real forms of the simple Lie groups, that's much easier and follows directly from the classification of the complex simple Lie algebras over $\mathbb{C}$, but, if you want all the real forms, you need to know the finite order automorphisms of the complex simple Lie algebras, which is more delicate. $\endgroup$
    – Robert Bryant
    Commented yesterday
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    $\begingroup$ @QGravity: This information about roots and fundamental weights can also be found in Helgason's book. $\endgroup$
    – Robert Bryant
    Commented 20 hours ago
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There is an abstract way of integrating Lie algebras but I guess you are asking for a more hands on approach. I suggest browsing Exceptional Lie groups by Ichiro Yokota. Usually, it's the compact (or perhaps complex) Lie groups which are treated in the literature, so maybe you should be more specific in what do you think is lacking there.

As far as $F_4$ goes the description can be made rather succinct using so called Jordan algebra:

Take the real vector space of octonionic-Hermitian three by three matrices and endow it with commutative product defined by $A \circ B = \frac{1}{2}(AB+BA).$ The automorphism group of this product is the compact Lie group of type $F_4.$ One can actually define some kind of octonionic determinant for this Jordan algebra and then it can be proved that $F_4$ is the group stabilizing this determinant and trace. If you take just the group stabilizing the determinant, you will obtain noncompact real form of $E_6.$

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  • $\begingroup$ Thank you. Well, I don't really know if it is lacking in the literature, since I don't know the literature very well, to be honest. Most of the books I have focus on Lie algebras rather than Lie groups, and when they do talk about Lie groups, they usually do something general, rather than something specific. These are great books though. $\endgroup$
    – Malkoun
    Commented Jun 21, 2020 at 21:54
  • $\begingroup$ However, as Weyl wrote in his "Classical Groups" book, though I cannot remember his exact words, even if one knows the general theory, that does not mean one should not work out the examples (he was talking about invariant theory, and Hilbert's theorems). $\endgroup$
    – Malkoun
    Commented Jun 21, 2020 at 21:56
  • $\begingroup$ Ideally, I would like to see the descriptions made by Cartan of the $5$ exceptional Lie groups but in a more modern way. I thank you for the octonionic description of $F_4$ and for the reference. I will check it out. $\endgroup$
    – Malkoun
    Commented Jun 21, 2020 at 21:59
  • $\begingroup$ I apologize for changing the accepted answer. I really thank you for the reference, which seems to take on an octonionic approach. For my purposes though, it is Cartan's work which is closer to what I am currently interested in. I apologize again. $\endgroup$
    – Malkoun
    Commented Jun 22, 2020 at 9:49
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    $\begingroup$ No worries. This site is more about math than about egos. ;) $\endgroup$
    – Vít Tuček
    Commented Jun 22, 2020 at 9:52

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