Exceptional Lie groups and algebras in maths and physics

Question

By the beautiful classification theorem of complex semisimple Lie algebras, we know that there are exactly $5$ types of exceptional Lie algebras, say type $E_6,E_7,E_8,F_4$ and $G_2$. We have a general approach to construct Lie algebras from the corresponding root systems. Howerer, I want to know some concrete and specific constructions. How did Killing and Dynkin do this? I've heard that it's related to Octonions $\mathbb{O}$.

Moreover, I know that exceptional Lie groups and algebras arise as symmetries for many modern physics structures in quantum field theory and string theory, are there any good references?

My background: I know standard theory of complex semisimple Lie algebras and differential geometry. Also I'm familiar with classical mechanics, and know some quantum mechanics. I can read Chinese and English, and both technical and popular literatures are welcome.

Thanks in advance.

Answer 1

There is a nice article by Alberto Elduque on Tits construction of the exceptional simple Lie algebras. Tit's found this construction $1966$, when Dynkin was $42$ years old.

On this site, there are several references, too. For $E_8$ I wrote some references in this post

Exceptional Lie algebras E8

For $G_2$ see also here:

Understanding $G_2$ as a particular subgroup of $SO(7)$

Also, the pages of John Baez here contain interesting informations. On the Lie group level, see also this MO-post:

Beautiful descriptions of exceptional groups

Answer 2

In addition to Dietrich's answer I have found the following flavour of constructions quite useful (although I don't have a good reference to a place looking at these in more detail) especially since many constructions focus on the compact story and my interest is more in the split/complex case.

Take $E_7$ as an example. It contains several maximal rank root subsystems e.g. $A_7, A_5 \times A_2, D_6 \times A_1, A_3 \times A_3 \times A_1$. You can see these by (recursively) extending the Dynkin diagram and deleting a node. Each of these correspond to a subalgebra of the same rank as $E_7$. So, for example, the $A_7$ subsystem describes a copy of $\mathfrak{sl}_8 \leq \mathfrak{e}_7$ spanned by certain root spaces and the Cartan subalgebra. We can then try to interpret $\mathfrak{e}_7$ via this classical Lie algebra. In fact the remaining root spaces must form a representation of $\mathfrak{sl}_8$. In this case this turns out to look like $\bigwedge^4 V$ where $V$ is the standard 8-dimensional representation of $\mathfrak{sl}_8$. So $\mathfrak{e}_7 \cong \mathfrak{sl}_8 \oplus \bigwedge^4 V$ where the bracket on the first summand is the usual bracket in $\mathfrak{sl}_8$, the bracket $[\mathfrak{sl}_8,\bigwedge^4 V]$ is given by the representation of $\mathfrak{sl}_8$ on $\bigwedge^4 V$. The bracket on $\bigwedge^4 V$ is a little more complicated but can be worked out by hand. A helpful observation is that root spaces in $\mathfrak{sl}_8$ look like $L_i^*\otimes L_j$ for $V= L_1 \oplus \cdots \oplus L_8$ and the root spaces in $\bigwedge^4 V$ are $L_i \wedge L_j \wedge L_k \wedge L_m$. Note these are exactly the two camps that the $E_7$ roots are often divided into with the latter corresponding to the roots of the form $\frac{1}{2}(\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4) - \frac{1}{2}(\epsilon_5+\epsilon_6+\epsilon_7+\epsilon_8)$

We can use this to understand the representations as well. The lowest dimensional representation of $\mathfrak{e}_7$ can be viewed as $ \bigwedge^2 V \oplus \bigwedge^2 V^* $ where the $\mathfrak{sl}_8$ acts as you'd expect preserving the summands and $\bigwedge^4 V$ swap them acting via $\bigwedge^4 V \wedge \bigwedge^2 V = \bigwedge^6 V \cong \bigwedge^2 V^*$ and $\bigwedge^4 V^* \wedge \bigwedge^2 V^* = \bigwedge^6 V^* \cong \bigwedge^2 V$ using the fact that $\bigwedge^k V \cong \bigwedge^{8-k} V^*$.

More detail on this specific construction can be found in this paper by Cacciatori, Dalla Piazza, and Scotti in section 4.

I have never gotten around to looking up the original papers but apparently Cartan used a construction of $\mathfrak{e}_6$ via a maximal subalgebra of the form $\mathfrak{sl}_6 \oplus \mathfrak{sl}_2$. Letting $V$ be the 6-dimensional representation and $W$ the 2-dimensional one, the other roots span a representation of the form $\bigwedge^3 V \otimes W$ and the lowest dimensional representations are $ \bigwedge^2 V^* \oplus (V \otimes W)$ and its dual. Credit to Robert Bryant for pointing out this last part to me on Math Overflow.