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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

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Some terminology needs translation: when you write "the Cartan Matrix becomes zero ...", I read it as a statement about the determinant of the matrix, etc. Lie theory originated in questions about mathematics and only gradually got taken up by various kinds of physicists overcoming fear of the "Gruppenpest". At first only the classical Lie groups were thought important, but in theoretical areas like string theory the exceptional ones have now caught on. Affine Kac-Moody algebras were at first just an abstract curiosity but then turned out to fit needs of conformal field theory. –  Jim Humphreys Jul 31 '10 at 13:44
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The invariant form needs not be symmetric (or hermitian symmetric, if compact Lie algebras are considered). You have skipped over the C series (symplectic groups), where the defining representation has an invariant skew-symmetric form. Any representation of a semisimple Lie algebra has an invariant bilinear form (the trace form). If the Lie algebra is simple and the representation is irreducible then the form is unique up to multiplication by a scalar and consequently either symmetric or skew-symmetric. Of course, in general there will be other invariants, which are hard to describe. –  Victor Protsak Jul 31 '10 at 17:49

6 Answers 6

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.

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The standard reference on these is the book "infinite dimensional Lie algebras" by V. Kac –  Richard Borcherds Jul 31 '10 at 13:42
    
Is there a standard reference for the last sentence of your answer? –  S. Carnahan Aug 1 '10 at 5:02
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Since the (say, complex) group E8 is adjoint and its Dynkin diagram has no outer automorphisms, it is the automorphism group of its Lie algebra -- its (nontrivial) 248-dimensional representation. But what form do you have in mind? The only one I see is the tensor that encodes the Lie bracket. Do you know something more interesting? –  Skip Aug 6 '10 at 18:26

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!

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Regarding E7: the stabilizer of the E7-invariant quartic form on the 56-dimensional representation has two components, of which the simply connected E7 is the identity component. The two components are distinguished by whether they preserve the E7-invariant skew-symmetric bilinear form or not. So to get just E7, you have to take the stabilizer of both the bilinear and the quartic form. –  Skip Aug 6 '10 at 18:14

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.

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+1. Indeed, $E_9$ is the affine Lie algebra $\hat{E}_8^{(1)}$. There is work by Louise Dolan and collaborators on this algebra, in the physics literature. –  José Figueroa-O'Farrill Jul 31 '10 at 13:35

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.

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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.

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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.

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It's Borcherds, by the way, not Borchers. –  José Figueroa-O'Farrill Aug 5 '10 at 13:29
    
@José Thanks, I have corrected it. –  JME Aug 5 '10 at 13:50
    
JSE, I am sorry to dampen your enthusiasm, but can you, please, explain how does your discussion of supergravity theory and string theory applications of some infinite-dimensional Kac-Moody algebras answer two specific questions formulated by the OP? It reads instead like a long blog entry laced with physics jargon. –  Victor Protsak Aug 5 '10 at 16:06
    
@VictorProtsak I think my answer is in the flow of the discussion following the other answers. I am not sure what you call "physics jargon" and I am deeply sorry if you feel uncomfortable with it, but the person who asked the question mention that his background is theoretical physics. –  JME Aug 5 '10 at 16:37
    
By "physics jargon" I mean passages like this one: $$ $$ These massive supergravity theories are obtained by gauging a group of isometry of the scalar manifold, allowing non-exact p-forms (turning on fluxes) or by adding a cosmological constant like in the case of type IIA supergravity with the Romans mass. They can often be understood in terms of branes that couple to the p-forms or their duals. $$ $$ –  Victor Protsak Aug 5 '10 at 18:59

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