Exceptional Lie algebras E8

Question

I have some questions about the exceptional Lie algebras, in particular on Lie algebra E8

1) Wath difference from the other Lie algebras, especially the classic Lie algebras $A_{l}$, $B_{l}$,$C_{l}$ and $D_{l}$?

2) Exists other Lie algebras, for example Lie algebras E9, E10, etc.?

3) All theory related to Lie algebras, also applies to exceptional Lie algebras or where it fails?

4) Exits open questions about Lie algebra E8, can that research?

5) Finally, what the possible applications of Lie algebra E8 ?

Thank you very much for your help

Answer 1

1) A simple Lie algebra of type $E_8$ is different from simple Lie algebras of classical type, because it does not fit into any of these series. That's the reason for the name "exceptional" Lie algebra. Otherwise a simple Lie algebra of type $E_8$ is different from all Lie algebras, which are not simple.

2) The classification of root systems shows that there are no finite-dimensional simple Lie algebras of type $E_n$ for $n>8$. If you allow infinite dimensions, however, there can be said more - see this question.

3) All theory which can be applied to Lie algebras, of course then also applies to a Lie algebra of type $E_8$. However, not all properties of classical Lie algebras are shared by the exceptional ones.

4) Yes, there are several open questions concerning Lie algebras of type $E_8$ in research. You will certainly find many papers during a search. I can advertise our paper on Etale representations for reductive algebraic groups with factors $Sp_n$ or $SO_n$, where we ask ourselves, which Lie algebras $\mathbb{C}\oplus \mathfrak{s}_1\oplus \cdots \oplus \mathfrak{s}_n$ admit a pre-Lie algebra structure (directly related to etale representation), e.g., can one of the simple factors be of type $E_8$? For details see section $5$ in this paper.

5) See section $9$ here.

Answer 2

The answer of Dietrich Burde is excellent, so I'm just adding some remarks about $E_8$ and connexion with algebraic geometry.

There is a fascinating relation with finite subgroups of $\rm{SU}_2$, Du Val singularities and simple Lie algebra. More precisely all these objects up to isomorphisms are classified by the Dynkin diagram. In this setting, the group corresponding to $E_8$ is the binary icosahedral group $\rm{BI}$, of order $120$, which can be descibed as follows : there is a double cover $\rm{SU}_2 \to \rm{PSU}_2 \cong \rm{SO}_3$ (corresponding to the double cover $S^3 \to \Bbb RP^3$). There is a subgroup $G \subset \rm{SO}_3$ with $G \cong \mathfrak A_5$, corresponding to the isometry group of a icosahedron. Then, we take ${\rm{BI}} = \pi^{-1}(G)$. The corresponding singularity is given by $x^2 + y^3 + z^5 = 0$. This was already know by Klein !

Finally, here is a mysterious connexion between Lie algebras of type $E$ and algebraic geometry : let $X$ be a smooth cubic surface in $\Bbb P^3$, then it is well known that there is $27$ lines on $X$. The smallest representation of $E_6$ is of dimension $27$, corresponding to the lines of a cubic surface !

But that's not all. We can project our cubic surface $X$ to a plane, this will realize $X$ as a branched cover of $\Bbb P^2$ over a quartic $Q$. Every line $L \subset X$ will be send into a line bitangent to $Q$. Moreover, the projection being not defined everywhere, we need to perform a blow-up for make this map well-defined, which gives a $28$-th bitangent. The fundamental representation of $E_7$ has dimension $56 = 2 \cdot 28$. I don't know the connexion but it seems plausible at least.

But according to this philosophy, $E_8$ should be related to tritangent planes of a sextic $C \subset \Bbb P^3$ which is the intersection of a quadric and a cubic surface. I have no ideas how these two are related.