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I am interested in the details of Elie Cartan's thesis, and, more specifically the explicit construction of the exceptional Lie groups as groups of symmetries of some specific homogeneous polynomials (according to what I have read in many places). I am interested in the details. For instance, what does one such a polynomial look like? What do all such polynomials (with groups of symmetry a fixed exceptional Lie group) look like? Is the set of all such polynomials dense in the relevant polynomial space?

Also, I am interested in different realizations of the exceptional Lie groups: for instance, $G_2$ is the group of symmetries of a skew-symmetric trilinear form on $\mathbb{C}^7$. And in this case, the space of such skew-symmetric trilinear forms is dense in the set of all skew-symmetric trilinear forms on $\mathbb{C}^7$.

So I guess in general, I am interested in concrete and explicit realizations of the exceptional Lie groups as groups of symmetries of a specific algebraic object. Also what is the set of all such objects with this property? And, is the set of all such objects dense in the relevant vector space? I hope my questions are sufficiently clear!

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    $\begingroup$ Have you checked directly his thesis archive.org/details/surlastructured00bourgoog? $\endgroup$ Oct 19 '16 at 8:16
  • $\begingroup$ Thank you so much!! I did not know where to find it online. I will have a look. $\endgroup$
    – Malkoun
    Oct 19 '16 at 8:21
  • $\begingroup$ Your welcome. His complete publications were also edited, although I think that they are only available in paper form. I hope you have access to a good library! $\endgroup$ Oct 19 '16 at 8:28
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    $\begingroup$ As Paul-Benjamin indicates, the collected papers of E. Cartan take up four hefty volumes (arranged in three parts) in the 1984 CNRS edition. Here the long thesis is number 5 out of the 186 publications listed. It occurs in Partie I "Groupes de Lie" (1356 pages!), or more conveniently now in the online link given.. $\endgroup$ Oct 19 '16 at 14:04
  • $\begingroup$ Thank you Prof. Humphreys. Yes, it would take a while to read all his work! I will not attempt that, but it would be really nice to read a selection of Elie Cartan's work. In some sense, "history" only retains part of each great mathematician's work, but when one reads the sources, one realizes there is much more in their work than one might get the impression a priori. $\endgroup$
    – Malkoun
    Oct 19 '16 at 14:07
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$\mathrm{G}_2$ is the only one of the exceptional groups that can be defined as the stabilizer of a `generic' tensorial object on a vector space and, over the complex numbers, even this is not quite right.

More precisely, let $V$ be a vector space (over $\mathbb{F}$, which could be $\mathbb{R}$ or $\mathbb{C}$) of dimension $d$ and let $\mathrm{Sch}(V)$ be some Schur functor applied to $V$. For example, $\mathrm{Sch}(V)$ might be $S^k(V^*)$, the symmetric homogeneous polynomials of degree $k$, or $\Lambda^k(V)$, the exterior $k$-forms of degree $k$. Or it might be something less standard, such as the kernel of the natural map $$ W:\Lambda^k(V)\otimes V \longrightarrow \Lambda^{k+1}(V) $$ or the natural trace map $$ \mathrm{tr}: V\otimes V^* \to \mathbb{F}. $$

It is very rare that $\mathrm{GL}(V)$ acts on $\mathrm{Sch}(V)$ with open orbits outside of the classical cases, such as $$ \mathrm{Sch}(V) = V,\ V^*,\ S^2(V),\ S^2(V^*), \Lambda^2(V), \Lambda^2(V^*) $$ or these tensored with some power of the $1$-dimensional representations $\Lambda^d(V)$ or $\Lambda^d(V^*)$. When $d=2$, you also have $S^3(V)$ and $S^3(V^*)$, and when $d=6,7,8$, you have $\Lambda^3(V)$ and $\Lambda^3(V^*)$ (and, of course, these tensored with some power of the $1$-dimensional representations $\Lambda^d(V)$ or $\Lambda^d(V^*)$). That's about it.

When $\mathrm{GL}(V)$ acts on $\mathrm{Sch}(V)$ with an open orbit $\mathcal{O}\subset\mathrm{Sch}(V)$, we say that an element $w\in\mathcal{O}$ is stable. The $\mathrm{GL}(V)$-stabilizer of such a stable element $w$ is the group $G_w\subset\mathrm{GL}(V)$.

In the classical cases, these stabilizers are essentially the orthogonal and symplectic groups. In the case $d=7$ and $\mathrm{Sch}(V)=\Lambda^3(V^*)$ or $\Lambda^3(V)$ (or these twisted by some power of $\Lambda^7(V)$ or $\Lambda^7(V^*)$), the identity component of the stabilizer of a stable element is $\mathrm{G}_2$. These are the only cases in which the stabilizer of a stable element is (up to finite extension) an exceptional group.

The other exceptional groups $G\subset \mathrm{GL}(V)$ do occur as stabilizers of non-stable elements of some $\mathrm{Sch}(V)$. For example, when $d=26$, there is an element of $S^3(V^*)$ whose stabilizer has identity component $\mathrm{F}_4$, when $d=27$, there is an element of $S^3(V^*)$ whose stabilizer has identity component $\mathrm{E}_6$, when $d=56$ there is an element of $S^4(V^*)$ whose stabilizer has identity component $\mathrm{E}_7$, and when $d=248$ there is an element of $\Lambda^3(V^*)$ whose stabilizer has identity component $\mathrm{E}_8$. None of these are stable elements, though.

(These are the answers when $\mathrm{F}=\mathrm{C}$. The story in the case of real forms is more complicated. For that, you should look in Cartan's 1913 paper classifying the real forms of the complex simple groups.)

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  • $\begingroup$ Thank you so much Prof. Bryant. Great answer! I have another question, but, since it is different from this one, though related, I think it is better to create another post. $\endgroup$
    – Malkoun
    Oct 19 '16 at 9:53
  • $\begingroup$ Here is a link to Cartan's 1914 paper on real forms of simple groups mentioned by Robert. $\endgroup$ Oct 21 '16 at 19:53
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All details about Cartan's thesis can be found in the thesis itself:

https://archive.org/details/surlastructured00bourgoog

There is also a German translation for those who do not read French:

Ueber die einfachen Transformationsgruppen. (German) JFM 25.0638.01 Leipz. Ber. XLV. 395-420 (1893).

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  • $\begingroup$ I think it's a bit of a stretch to say that all details about Cartan's thesis can be found in the thesis itself, unless you mean the tautological definition that a statement is a detail of Cartan's thesis if and only if it appears in Cartan's thesis. For example, Cartan does not explicitly verify in his thesis that the Jacobi identity holds for the exceptional algebras he writes down, though I'm sure he knew that it did. He also does not give any detail showing how his cubic in 27 variables defines $E_6$ or how he knows that its projectivized singular locus has dimension 16, for example. $\endgroup$ Oct 19 '16 at 19:42
  • $\begingroup$ @Robert Bryant: Sure, I meant this in tautological sense:-) I have never read this thesis myself, reading instead Bourbaki volume, and other books. $\endgroup$ Nov 4 '16 at 2:38

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