## October 4, 2015

### Configurations of Lines and Models of Lie Algebras (Part 1)

#### Posted by John Baez

I’m really enjoying this article, so I’d like to talk about it here at the n-Category Café:

It’s a bit intense, so it may take a series of posts, but let me just get started…

I started reading this paper because I wanted to finally understand the famous “27 lines on a cubic surface” and how they’re related to the smallest nontrivial representations of $\mathrm{E}_6$, which are 27-dimensional. Actually $\mathrm{E}_6$ has two nontrivial representations of this dimension, which are not isomorphic: one is the exceptional Jordan algebra, and one is its dual! The exceptional Jordan algebra consists of $3 \times 3$ self-adjoint octonionic matrices, so it has dimension

$8+8+8+3 = 27$

The determinant of such a matrix turns out to be well-defined despite the noncommutativity and nonassociativity of the octonions. $\mathrm{E}_6$ is the group of linear transformations of the exceptional Jordan algebra that preserves the determinant.

As you might expect, all this stuff going on in dimension 27 is just the tip of an iceberg—and Manivel explores quite a large chunk of that iceberg. But today let me just touch on the tip.

For starters, the Cayley–Salmon theorem says that every smooth cubic surface in $\mathbb{C}\mathrm{P}^3$ has exactly 27 lines on it.

(This is apparently one of those cases where mathematicians shared credit with the meal that inspired their work, like the Fermi–Pasta–Ulam problem.)

I can’t visualize those 27 lines in general. But Clebsch gave an example of a smooth real cubic surface where all the lines actually lie in $\mathbb{R}\mathrm{P}^3$, so you can see them. This is called the Clebsch surface, or Klein’s icosahedral cubic surface, because Klein also worked on it. It looks like this:

and the 27 lines look like this:

Please click on the pictures to see who created them! Here’s a model of it — one of those nice old plaster models you see in old universities:

This model is in Göttingen, photographed by Oliver Zauzig.

I would enjoy diving down the rabbit hole here and learning everything about this particular cubic surface, but I’ll resist for now! I’ll just say a few things:

First, the Clebsch surface can be described very nicely as a surface in $\mathbb{R}\mathrm{P}^4$ using the homogeneous equations

$x_0+x_1+x_2+x_3+x_4 = 0$ $x_0^3+x_1^3+x_2^3+x_3^3+x_4^3 = 0$

but then you can eliminate one variable and think of it as a surface in $\mathbb{R}\mathrm{P}^3$ given by the equation

$x_1^3+x_2^3+x_3^3+x_4^3 = (x_1+x_2+x_3+x_4)^3$

Second, the lines are actually defined over the golden field $\mathbb{Q}[\sqrt{5}]$. This may have something to do with why it’s called ‘Klein’s icosahedral cubic surface’ — I’ll avoid looking into that right now, but it may eventually be important, because there are nice relations between some exceptional Lie algebras and the golden field.

Third, you can see some points where three lines intersect: these are called Eckardt points and there are 10 of them.

Anyway, you may be wondering why there are 27 lines on a smooth cubic surface. The best argument I’ve seen so far, in terms of maximum friendliness, minimum jargon, and maximum total insight conveyed, is here:

I can’t say I fully understand it, since it’s fairly involved, but I still recommend it to anyone who knows a reasonable amount of algebraic geometry.

Anyway, I don’t think one needs to fully understand this to start wondering what $\mathrm{E}_6$ has to do with it. Here’s some of what Manivel has to say:

The configuration of the 27 lines on a smooth cubic surface in $\mathbb{C}\mathrm{P}^3$ has been thoroughly investigated by the classical algebraic geometers. It has been known for a long time that the automorphism group of this configuration can be identified with the Weyl group of the root system of type $\mathrm{E}_6$, of order 51,840. Moreover, the minimal representation $J$ of the simply connected complex Lie group of type $\mathrm{E}_6$ has dimension 27.

Here the letter $J$ means ‘exceptional Jordan algebra’.

This is a minuscule representation, meaning that the weight spaces are lines and that the Weyl group $W(\mathrm{E}_6)$ acts transitively on the weights. In fact one can recover the lines configuration of the cubic surface by defining two weights to be incident if they are not orthogonal with respect to the unique (up to scale) invariant scalar product. Conversely, one can recover the action of the Lie group $\mathrm{E}_6$ on $J$ from the line configuration.

I hope to come back to this and keep digging deeper. Right now I don’t even understand how the 27 weight spaces in $J$, which are 1-dimensional subspaces in a 27-dimensional space, are connected to 27 lines in some surface. But there are some other things in Manivel’s paper that I understand and like a lot.

Posted at October 4, 2015 11:06 PM UTC

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### Re: Configurations of Lines and Models of Lie Algebras (Part 1)

Incidentally, it’s very rare in some sense to have representations not be isomorphic to their duals. Dualizing gives an involution of the Dynkin diagram (since its vertices correspond to fundamental representations), and most Dynkin diagrams don’t have involutions. Also, $D_4$ has involutions but no natural one, and indeed $D_{even}$ has none of these non-self-dual representations.

The only groups that have them are $A_{(n \gt 1)}$, $D_{odd}$, and $E_6$.

I’m guessing that if there’s a satisfying reason for this 27 to be the dimension of the $E_6$ rep, it’ll be via del Pezzo surfaces and will work for $E_6$, $E_7$, $E_8$, and $E_5 = D_5$, $E_4 = D_4$, $E_3 = A_3$, $E_ 2 = A_1 \times A_1$, and maybe even half-work for

Posted by: Allen K. on October 5, 2015 1:43 AM | Permalink | Reply to this

### Re: Configurations of Lines and Models of Lie Algebras (Part 1)

Del Pezzo surfaces are indeed part of the story. I don’t understand anything about them, but Manivel’s paper starts like this:

Such classical configurations of lines as the 27 lines on a complex cubic surface or the 28 bitangents to a smooth quartic plane curve have been extensively studied in the 19th century (see e.g. [19]). Their automorphism groups were known, but only at the beginning of the 20th century were their close connections with the Weyl groups of the root systems $\mathrm{E}_6$ and $\mathrm{E}_7$ recognized, in particular through the relation with Del Pezzo surfaces of degree three and two, respectively [9,10]. Del Pezzo surfaces of degree one provide a similar identification of the diameters of the root system $\mathrm{E}_8$, with the 120 tritangent planes to a canonical space curve of genus 4.

Posted by: John Baez on October 5, 2015 1:57 AM | Permalink | Reply to this

### Re: Configurations of Lines and Models of Lie Algebras (Part 1)

It seems a good time to revisit Mysterious Duality, where $E_6$ corresponds to the del Pezzo $B_6$. In the paper we find toric geometric descriptions of the del Pezzo surfaces $B_k$, for k<=3. For k>=4, $B_k$ does not admit a toric description. However, for k=4,5,6,7,8 we find the gauge groups $SU(5)$, $SO(10)$, $E_6$, $E_7$, $E_8$, which are very popular GUT groups.

The toric geometric descriptions of $B_k$, for k=1,2,3 are merely CP^2 blown up at 1, 2, and 3 points, respectively. For anyone with a grasp of the Jordan algebraic description of CP^2, such constructions are especially interesting, given that such points means “blowing up” independent projectors of the Jordan algebra. Given one is restricted to a world devoid of quaternions and octonions, the complex variable toric geometry indeed terminates at three blow ups. However, a lover of division algebras could imagine generalizing such a toric description, where instead of considering circle fibrations over intervals (giving CP^1’s), we consider 7-sphere fibrations over intervals (giving OP^1’s~8-spheres). We could then study OP^2 with k=1,2,3 blow ups, at which we recover three exceptional curves and the process terminates. We know OP^2 with no blowups has a natural $F_4$ symmetry. So what of OP^2 with blowups at k=1,2,3? Do these have any relation to $E_6$, $E_7$ and $E_8$?

Posted by: Metatron on October 5, 2015 2:28 PM | Permalink | Reply to this

### Re: Configurations of Lines and Models of Lie Algebras (Part 1)

To be more precise, let’s define $\mathbb{KB}_k$ to be the projective plane over division algebra $\mathbb{K}$ with k blown up points. Then my conjecture is that the following relations hold:

1) $\mathbb{RB}_3\sim \mathbb{CP}^2$

2) $\mathbb{CB}_3\sim \mathbb{HP}^2$

3) $\mathbb{HB}_3\sim \mathbb{OP}^2$

4) $\mathbb{OB}_3\sim (\mathbb{C}\otimes\mathbb{O})\mathbb{P}^2$

$E_6(\mathbb{C})$ acts on the complexified Cayley plane in the usual fashion. It remains to relate this to the del Pezzo $\mathbb{B}_6$, which may be the projective plane over the sextonions $\mathbb{SP}^2$ in disguise.

Posted by: Metatron on October 5, 2015 5:50 PM | Permalink | Reply to this

### Re: Configurations of Lines and Models of Lie Algebras (Part 1)

Dear John,

I know that

For starters, the Cayley–Salmon theorem says that every smooth cubic surface in ℂP3 has exactly 27 lines on it.

(This is apparently one of those cases where mathematicians shared credit with the meal that inspired their work, like the Fermi–Pasta–Ulam problem.)

was a joke, but I think it does a disservice to Salmon, who was an important figure in the development of algebraic geometry in the mid-19th century.

In particular, his book A treatise on the analytic geometry of three dimensions was, if I understand correctly, the standard reference on this stuff for a long period. (Unfortunately after that he switched his interest from math to theology, and followed it up with the less crowd-pleasing The eternity of future punishment.)

Posted by: Artie Prendergast-Smith on February 5, 2016 9:34 AM | Permalink | Reply to this

### Re: Configurations of Lines and Models of Lie Algebras (Part 1)

I think anyone who writes a book called the The Eternity of Future Punishment should be able to take little posthumous ribbing. But no dishonor was meant to George Salmon — nor John Pasta, for that matter!

Posted by: John Baez on February 6, 2016 6:44 AM | Permalink | Reply to this

### Re: Configurations of Lines and Models of Lie Algebras (Part 1)

Fair enough. I think I am a little defensive of Salmon because he is the unique famous Irish algebraic geometer, but I guess as you say he probably has bigger worries at the moment (if his book is correct).

Ironically, there is a “Salmon conjecture” in algebraic geometry, but that is named after an actual fish.

Posted by: Artie Prendergast-Smith on February 6, 2016 9:39 AM | Permalink | Reply to this

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