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Results tagged with exceptional-groups
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user 13972
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.
3
votes
Embed exceptional non-compact simply connected simple Lie groups into classical simple Lie g...
This fails already for the split form of $\mathrm{G}_2$. Every finite-dimensional irreducible representation of its Lie algebra is a constituent of a tensor power of the $7$-dimensional representation …
- 108k
7
votes
Accepted
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{ …
- 108k
5
votes
Diagonalization of octonionic Hermitian matrices of size $2\times 2$
Yes, in fact, any $2$-by-$2$ octonionic Hermitian matrix is equivalent under the natural $\mathrm{Spin}(9)$ action to a diagonal $2$-by-$2$ octonionic Hermitian matrix.
This follows from the well-know …
- 108k
7
votes
Accepted
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 …
- 108k
7
votes
Accepted
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 …
- 108k
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 …
- 108k
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 …
- 108k
6
votes
Accepted
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), …
- 108k
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 …
- 108k
4
votes
Accepted
A representation of Spin(9,1)
In this case if your 16-dimensional $\mathrm{Spin}(9,1)$-representation is the one of highest weight $(0,0,0,0,1)$, then the $\mathrm{Spin}(9,1)$-irreducible decomposition of its symmetric square is j …
- 108k
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 $\mathrm{SO}(9)$, …
- 108k
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 …
- 108k
5
votes
Accepted
$Spin(7)$ 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 …
- 108k
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 …
- 108k
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 …
- 108k