TITLE: Exceptional Lie algebras 
QUESTION [8 upvotes]: I have some questions regarding the exceptional Lie algebras e(n), n=6,7,8.
Can anybody explain to me what prevents us from constructing e(9) from e(8)? One can use the e(8) lattice vectors and try to construct an e(9) vector; one could go even further and try e(10) etc. I know that the Cartan Matrix becomes zero (or negative for 10, ...) which is forbidden, but what does that mean if one would try to write down the generators for e(9)? What's wrong with them as Lie algebra generators? Where does the re-construction of the Lie algebra from the Dynkin diagram / Cartan matrix fail?
Another question I have is related to E(n) as symmetry groups. For the A, B, C and D series one can understand the (fundamental or defining representation of) Lie groups acting on a certain vector space and leaving a certain scalar product invariant. For SO(n) it's (x,y) with x,y living in a real vector space, for SU(n) it's (x*, y) with x,y living in a complex vector space. What about E(n)? Is there a similar scalar product which is invariant? What is the corresponding vector space? Are there other invariants?
(my background is theoretical physics, particle physics, gauge symmetries etc.)
Thanks
Tom

REPLY [3 votes]: The appearance of exceptional Lie algebra, Kac-Moody algebras and Borcherds algebra in gravitational theories is a very elegant  and exciting corner of current research in supergravity and string theory. I would like to discuss how exceptional Lie algebras naturally appear in the context of maximal supergravity theories and how this is connected to $E_{11}$ and Borcherds superalgebras.  I will also discuss how Kac-Moody algebra  occurs naturally in the context of gravitational singularities. 
11 dimensional supergravity and exceptional Lie algebra
11 dimensional supergravity was constructed in 1978 by  Cremmer, Julia and Scherk. 
Nowadays it is considered as the low energy limit of M-theory. 
The  field content of 11-dimensional supergravity  is simply given by a metric $g$ and a 3-form $A_{(3)}$. 
 We can construct all the (massless)
 maximal supergravity theory in
 $D$-dimension with $2 < D <11$, by
 considering  (Kaluza-Klein) reduction
 of 11 dimensional supergravity on a
 $(11-D)$-torus.  This process produces a lot of additional fields (2-forms, 1-forms and 0-forms) coming from the reduction of the metric and the 3-form. 
 In general for a compactification of 11 dimensional supergravity on
 a torus $T^{11-D}$ to a D-dimensional
 spacetime, we produce a scalar
 manifold   $$
 \frac{E_{11-D}}{K(E_{11-D})},$$ 
where $K(G)$ is the maximal 
 compact subalgebra of $G$.  In particular, we have in dimension 5,
 4 and 3
$$ 5D\rightarrow \frac{E_6}{USp(8)},
 \quad 4D\rightarrow \frac{E_7}{SU(8)},
 \quad 3D\rightarrow
 \frac{E_8}{SO(16)}.  $$
$E_{11}$ conjecture and Borcherds algebras
We recall that $E_9=E_8^+$ is understood as the extended Dynkin diagram of $E_8$. In the same way $E_{10}=E_8^{++}$ and $E_{11}=E_8^{+++}$ are the over-extended  and the very-extended Dynkin diagram of $E_8$.
 There is a conjecture introduced by
 Peter West in 2001 and supported by
 several facts that the Kac-Moody
 algebra $E_{11}$ is related to a 
 non-linear realization of M-theory and
 that $E_{11}$ can provide an 11
 dimensional origin not only of all
 massless maximal supergravity theories
 (including type IIB) but also of the
  massive ones.
A beautiful duality was discovered by Iqbal, Neitzke and Vafa between compactifications of M-theory on tori and  the second cohomology of some associated del Pezzo surfaces.  Now the full cohomology of theses surfaces spans the root lattice of a Borcherds superalgebra.
 Henry-Labordere Julia and Paulot have
 shown that some truncations of these
 Borchers algebras provide a
 classification of $p$-forms coming from
 tori reduction of (massive) maximal
 supergravity.  This classification matches the one of the $E_{11}$ conjecture of Peter West. The Borcherds description was recently proven to be systematically derived from the split real form of $E_{11}$ by Henneaux, Julia and Levie. 
Space-time singularities, Kac-Moody algebra and Cosmic billiards
A fascinating and non-speculative occurrence of $E_9$ and $E_{10}$ in a theory of gravity occurs when studying the behavior of gravity near a spacetime singularity. 
Belinskii, Khalatnikov and Lifchitz (BKL) have studied in details the general solution of Einstein equations near a spacetime singularity. As one reaches the singularity, the Einstein equations admit a chaotic behavior in time. Chitre and Misner has reformulated the BKL analysis in terms of a billiard motion in a 2 dimensional hyperbolic space. 
In higher dimension, the chaotic behavior disappear in spacetime dimensions greater than 10. 
In particular, in 11 dimensions, there is no chaos at all. But if one add a 3-form (like the one of 11 dimensional supergravity), chaos comes back. In higher dimension one can also describe the chaotic behavior by a billiard in a higher dimensional hyperboloic space. 
When a theory admits a
 compactification to three dimensions
 on a higher dimensional torus such
 that in the reduced 3 dimensional
 theory, the Lagrangian is given by
 Einstein-Hilbert action and a sigma
 model with target space a $G/H$  such
 that $G$ is a simple Lie group and $H$
 its maximal compact subgroup, the
 billiard table is a Coxeter polyhedron
 and the billiard group is a Coxeter
 group. The table billiard can be
 described by the over-extended
 Kac-Moody algebra $G^{++}$ associated
 with the group $G$.
 In particular $$ \text{The billiard
 associated with eleven supergravity is
 } E_8^{++}=E_{10}.$$
One can formulate the billiard dynamics as a motion in the Cartan subalgebra of the Kac-Moody algebra.