Exceptional Lie algebras

Question

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

Answer 1

See http://en.wikipedia.org/wiki/En_%28Lie_algebra%29

Nothing goes wrong when you construct a Lie algebra E10, E11, ... by generators and relations from the Cartan matrix. The only difference is that the Lie algebras you get are infinite dimensional. E9 is a central extension of the affine E8 algebra, but E10 and beyond seem rather a mess.

E6, E7, E8 can be represented as symmetry groups of various forms on spaces of dimensions 27, 56, and 248. E9 and beyond act naturally on infinite dimensional vertex algebras, but apart from E9 are not the full symmetry groups of these algebras.

Answer 2

I think your first question has been answered by Jim and Richard Borcherds, so perhaps I can add the following.

As Robin points out in his answer, exceptional Lie algebras are not classical and hence not characterised by leaving invariant a bilinear form, but they do, of course, have invariants in many representations. A consequence of the fact he mentions concerning $G_2$, is that $G_2$ can be characterised as leaving invariant a stable 3-form in $\mathbb{R}^7$.

Similarly, $E_6$ can be characterised as the stabiliser of a symmetric rank-3 tensor in the fundamental 27-dimensional representation. Similarly, $E_7$ leaves invariant a symmetric rank-4 tensor in the fundamental 56-dimensional representation, but I am not sure right now if this is a characterisation. This is related to Jordan algebras.

Finally, besides the appearance of $E_n$ in the context of string/M-/conformal field theory, there are at least two other places where $E_6$ and $E_7$ appear in theoretical physics.

For many years, $E_6$ was a possible candidate for a GUT gauge group since there was a way to fit the known spectrum in the standard model (and then some). I think that $E_7$ and $E_8$ had also been considered, but perhaps less seriously.

Finally, certain real forms of $E_6$ and $E_7$ do appear when discussing compactifications of eleven-dimensional supergravity down to five and four dimensions, respectively. For example, compactifying on $S^7$ gives rise to a gauged supergravity with 70 scalar fields belonging to the noncompact symmetric space $E_7/\mathrm{SU}(8)$, where $SU(8)\subset E_7$ is the maximal compact subgroup for this real form of $E_7$. A similar story happens in five dimensions, but there it is a real form of $E_6$ which appears and the scalars parameterise the noncompact symmetric space $E_6/\mathrm{Sp}(4)$, with $\mathrm{Sp}(4)$ again the maximal compact subgroup. This is explained in papers of Cremmer and Julia from 1978-9.

Curiously these two symmetric spaces are the only two symmetric spaces which according to the wikipedia page have no geometric interpretation! One more reason, as if more were necessary, to get geometers interested in supergravity!

Answer 3

Like most other mathematicians, I am not an expert on the mathematical physics literature related to Lie algebras. But the E series has led further into Kac-Moody algebras: affine, hyperbolic, ... The hyperbolic Kac-Moody algebra $E_{10}$ shows up quite a bit in the literature (and there is even $E_{11}$). For instance, a random search of MathSciNet turns up such papers as:

MR1354261 (96g:17028) 17B67 (17B69 17B81 81R10). Gebert, R.W. (D-HAMB-2P); Nicolai, H. (D-HAMB-2P), On E10 and the DDF construction. Comm. Math. Phys. 172 (1995), no. 3, 571–622.

MR1894911 (2003i:83107) 83E50 (17B81 81R10 81T30 83E30). West, P. (4-LNDKC), E11 and M theory. Classical Quantum Gravity 18 (2001), no. 21, 4443–4460.

Mathematicians like Victor Kac and Ed Frenkel have been active in some of this work involving Lie theory and physics. As others have pointed out, the exceptional finite dimensional Lie algebras don't arise directly from quadratic forms and such in the way classical Lie algebras do, but they do come up in "symmetry" questions related to physics (at least theoretically) and do have interesting characterizations in terms of exceptional algebraic structures like octonions and the 27-dimensional special Jordan algebra. Similarly, some of the infinite dimensional Kac-Moody analogues are natural for symmetry questions in modern physics.

Answer 4

I expect if you did construct a Lie algebra with relations built from a "forbidden" Cartan matrix then you would get an infinite-dimensional Kac-Moody algebra or something similar.

Also for the exceptional Lie algebras, these are not classical Lie algebras which can be characterised as leaving certain quadratic forms invariant, but they can be constructed in other ways. For instance the split real $G_2$ Lie algebra is the algebra of derivations of the octonions or Cayley numbers.

Answer 5

I am certainly no expert, but one answer to your second question is that one can think of the classical Lie groups / algebras as certain constructions on $\mathbb{R}, \mathbb{C}$, and $\mathbb{H}$ which work in all dimensions. A natural question from here is to ask whether those constructions can be extended to the octonions, and the answer is only in finitely many dimensions! This is explained in Section 4 of John Baez's excellent online article about the octonions, particularly when he discusses the magic square.

Answer 6

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.