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Exceptional Lie groups G2, F4, E6, E7, E8 of dimensions 14, 52, 78, 133, 248 were obtained as result of classification of simple Lie groups performed by Killing and Elie Cartan. The tool used in classification is Dynkin diagram and root system of vectors in Lie algebra of the group. The remaining Lie groups form four infinite families of transformations of n-dimensional space over real (odd and even), complex and quaternionic field.

15 votes
Accepted

Where can I find details of Elie Cartan's thesis?

$\mathrm{G}_2$ 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 r …
Robert Bryant's user avatar
15 votes

Triality of Spin(8)

In addition to the above answers involving spinors and/or octonions, you might be interested in Cartan's original construction of the triality automorphisms, which is very explicit and takes just a co …
Robert Bryant's user avatar
14 votes

Matrix representation for $F_4$

While there have been many people who have done this, the first person to do so was Élie Cartan, who wrote down a basis for the matrix algebra ${\frak{f}}_4\subset{\frak{so}}(26)$ in his 1894 thesis S …
Robert Bryant's user avatar
12 votes
Accepted

What is the largest subgroup of $GL^{+}(7,\mathbb{R})$ which smoothly retracts onto $G_2$?

There is no retraction of $\mathrm{SO}(7)$ onto $\mathrm{G}_2$. If such a retraction $\rho:\mathrm{SO}(7) \to \mathrm{G}_2$ existed, then the composition $$ \mathrm{G}_2 \hookrightarrow \mathrm{SO}(7 …
Robert Bryant's user avatar
11 votes
Accepted

$Spin(7)$ as stabilizer of a $4$-form

There are at least two sources for this: First, Harvey and Lawson, Calibrated geometries (Acta Math. 1982) proves this (i.e., that the stabilizer of $\Omega_0$ is isomorphic to the nontrivial double …
Robert Bryant's user avatar
10 votes
Accepted

smallest simplest $E_8$ -module

Cartan showed that the lowest dimensional (nontrivial) $E_8$-module is ${\frak{e}}_8$ itself, i.e., the adjoint representation, which has dimension $248$. The next smallest nontrivial irreducible mod …
Robert Bryant's user avatar
10 votes

Constructing $E_8$ from its branching to $A_8$

Of course, the original description of this branching is due to Élie Cartan, himself. For example, see Chapitre IX of his 1914 paper Les groupes réels simples, finis et continu (Annales scientifiques …
Robert Bryant's user avatar
7 votes
Accepted

Explicit generators of the Lie algebra $spin(9)$

There are various places where you can see this written down, but let me suggest some notes that I wrote about spinors in the low dimensions that includes what you want, assuming that you know somethi …
Robert Bryant's user avatar
7 votes

Does $\mathrm{E}_7/(\mathrm{SU}_8/(\mathbb{Z}/2))$ carry an almost complex structure?

I'm sorry if this is obvious to everyone, but I thought that it was worth mentioning: I don't know the answer to the question asked, but the answer to the easier question, "Does the space $\mathrm{E} …
Robert Bryant's user avatar
7 votes
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Viewing exceptional Lie algebras via the classical ones

Élie Cartan himself, recognized and used the following description of $\mathfrak{e}_6$: Let $V$ be a vector space of dimension $6$ and let $W$ be a vector space of dimension $2$. Then there is a vect …
Robert Bryant's user avatar
7 votes
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Quadratic forms on $\mathbb{R}^{16}$ coming from octonions

I'm revising my answer because whether the image of the map $j$ that the OP defines is equal to the $10$-dimensional subspace $H$ of $\mathrm{Sym}^2(\mathbb{R}^{16})$ that is invariant under $\mathrm{ …
Robert Bryant's user avatar
7 votes
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How to describe the compact real forms of the exceptional Lie groups as matrix groups?

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 a …
Robert Bryant's user avatar
6 votes
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A question on complex semisimple Lie groups and $(\mathbb{C}^2, \omega)$

I think that the kind of question you are asking is one that was treated by Dynkin back in the 1950s (see Semisimple subalgebras of semisimple Lie algebras. (Russian) Mat. Sbornik N.S. 30(72), (1952), …
Robert Bryant's user avatar
5 votes
Accepted

Decomposition into irreducible components of a representation of $Spin(9)$

This is easily computed via LiE: $Sym^2(\mathbb{R}^{16})$ breaks into three irreducible components: The trivial representation, i.e., $\mathbb{R}$, The standard representation of $\mathrm{SO}(9)$, …
Robert Bryant's user avatar
5 votes
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$Spin(7)$ as stabilizer of a $4$-form revisited

I think that you want to look at the relevant passages in Spin Geometry by Lawson and Michelsohn, particularly Chapter IV, Sections 9 and 10, where they explain in general how the square of a spinor c …
Robert Bryant's user avatar

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