Integral Octonions (Part 1)

July 23, 2013

Posted by John Baez

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This is not particularly about $n$ -categories, but it’s related — somehow — to John Huerta’s work on division algebras and Lie n-superalgebras, which are the $n$ -categorified versions of Lie superalgebras, which in turn are the supersymmetric versions of Lie algebras.

You see, some of the most exciting Lie $n$ -superalgebras are those extending the Lie algebra of the Lorentz group in 10 or 11 dimensions, and these can be built using the octonions. The reason is that 10d Minkowski spacetime can be identified with the space $\mathfrak{h}_2(\mathbb{O})$ consisting of $2 \times 2$ hermitian matrices with octonions as entries. Thanks to this, the corresponding Lorentz group, the identity component of $SO(9,1)$ , can be identified with $PSL(2,\mathbb{O})$ . It is somewhat tricky to define $PSL(2,\mathbb{O})$ , because the octonions are nonassociative, but it can be done, and this caps off a very nice sequence of isomorphisms:

$\begin{array}{ccl} PSL(2,\mathbb{R})& \cong &SO_0(2,1) \\ PSL(2,\mathbb{C})& \cong &SO_0(3,1) \\ PSL(2,\mathbb{H})& \cong &SO_0(5,1) \\ PSL(2,\mathbb{O})& \cong &SO_0(9,1) . \end{array}$

Now I’d like I describe how this interacts with a further marvelous fact: the existence of ‘octonionic integers’.

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You see, one of the great things about

$PSL(2,\mathbb{R}) \cong SO_0(2,1)$

is that it gives us two ways to think about the symmetry group of the hyperbolic plane. On the one hand, it consists of fractional linear transformations with real entries:

$z \mapsto \frac{a z + b}{c z + d} \quad a,b,c,d\in \mathbb{R}$

acting on the upper half-plane in $\mathbb{C}$ , which is one way to think about the hyperbolic plane. On the other hand, it’s the identity component of the Lorentz group of 3d spacetime, acting on the hyperboloid

$\{(t,x,y) \in \mathbb{R}^{3} : t^2 - x^2 - y^2 = 1 , t > 0\}$

which is another way to think about the hyperbolic plane.

But the really great thing about this is that because the field $\mathbb{R}$ contains the integers as a discrete subring, $PSL(2,\mathbb{R})$ contains the modular group $PSL(2,\mathbb{Z})$ as a discrete subgroup, acting on the hyperbolic plane with fundamental domains like this:

This connects the Lorentz group of 3d spacetime to number theory… thanks to things like the theory of modular forms. Part of the point is that $PSL(2,\mathbb{Z})$ is not just a discrete subgroup of $PSL(2,\mathbb{R})$ . It’s an arithmetic group! Very very roughly, this is like an algebraic group, but defined using some ring resembling the integers, rather than a field.

And here is where a further marvelous fact comes into play. The algebra of octonions $\mathbb{O}$ contains a (nonassociative!) discrete subring $\mathbf{O}$ , sometimes called the Cayley integers, Coxeter integers, or octavians. If you ignore the fact that you can multiply octonions, you can think of the octonions as forming an 8-dimensional real inner product space… and then the Cayley integers form a lattice in this vector space, which is, up to a rescaling, nothing other than the $\mathrm{E}_8$ root lattice!

So, each Cayley integer has 240 nearest neighbors, arranged in a pattern like this:

which you should imagine in 8 dimensions.

That’s already great. But the Cayley integers are closed under multiplication, and this makes even more magic happen!

For example, it lets us define an arithmetic subgroup $PSL(2,\mathbf{O})$ of the group

$PSL(2,\mathbb{O}) \cong SO_0(9,1)$

This acts on the 9-dimensional hyperbolic space

$\{x \in \mathbb{R}^{10} :\; x_0^2 - x_1^2 - \cdots - x_9^2 = 1 , \; \x_0 > 0\}$

Even better, this marvelous discrete group acting on 9d hyperbolic space is related to $\mathrm{E}_8$ . In fact, $PSL(2,\mathbf{O})$ is the even part of the Coxeter group $\mathrm{E}_{10}$ , which arises from a Coxeter diagram like that of $\mathrm{E}_8$ , but longer:

This group has one generator $s_i$ for each dot, with relations

$s_i^2 = 1$

for each dot,

$s_i s_j s_i = s_j s_i s_j$

for any pair of dots connected by an edge, and

$s_i s_j = s_j s_i$

for pairs not connected by an edge. Each of these generators acts as reflection across a hyperplane in 9-dimensional hyperbolic space. So, it’s called a hyperbolic Coxeter group. The subgroup generated by products of two generators is called the even part of this group, and it’s the same as $PSL_2(\mathbf{O})$ .

The same thing happens in a much simpler way for $PSL_2(\mathbb{Z})$ : it’s the even part of a hyperbolic Coxeter group acting on the hyperbolic plane with fundamental domains like this:

You get these by taking fundamental domains for $PSL_2(\mathbb{Z})$ , which I showed you before in the upper half-plane, and chopping them in half.

So, thanks to Cayley integers, we know there is a similar but much more glorious picture in 9-dimensional hyperbolic space! Unfortunately this blog is too low-dimensional to contain it.

This Coxeter diagram:

can also be seen as a Dynkin diagram—of a dangerously big kind, since it gives an infinite-dimensional Lie algebra, a hyperbolic Kac–Moody algebra. This Lie algebra also goes by the name of $\mathrm{E}_{10}$ , and it’s closely related to the Coxeter group I just described: the Coxeter group is the Weyl group of the Lie algebra.

The Lie algebra $\mathrm{E}_{10}$ has tantalizing and mysterious connections to physics. Ideally, I would explain something about them in this series. At the very least, I want to say how hyperbolic Coxeter groups like $\mathrm{E}_{10}$ are related to supersymmetric quantum cosmological billiard balls! But the paper I want to recommend to you today:

• Axel Kleinschmidt, Hermann Nicolai, Jakob Palmkvist, Hyperbolic Weyl groups and the four normed division algebras.

has somewhat less cosmic goals, so it’s a good place to start. It describes how the isomorphisms

$\begin{array}{ccl} PSL(2,\mathbb{R})& \cong &SO_0(2,1) \\ PSL(2,\mathbb{C})& \cong &SO_0(3,1) \\ PSL(2,\mathbb{H})& \cong &SO_0(5,1) \\ PSL(2,\mathbb{O})& \cong &SO_0(9,1) . \end{array}$

work, with an emphasis on the subtleties introduced by noncommutativity (for the quaternions $\mathbb{H}$ and octonions $\mathbb{O}$ ) and nonassociativity (for $\mathbb{O}$ ). It describes various subrings of ‘integers’ in $\mathbb{R}, \mathbb{C}, \mathbb{H}$ and $\mathbb{O}$ and how these give rise to arithmetic groups. And, it describes how these are related to hyperbolic Coxeter groups. This is tremendously pretty math even if you don’t care about the potential applications to (fun, far-out, highly theoretical, not-yet-experimentally-testable) physics.

Posted at July 23, 2013 3:19 PM UTC

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7 Comments & 1 Trackback

Re: Integral Octonions (Part 1)

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I think in the sentence beginning ‘The subgroup generated by products of two generators is called the even part of this group’ you used the wrong font O at the end.

Posted by: Scott Morrison on July 25, 2013 2:41 AM | Permalink | Reply to this

Re: Integral Octonions (Part 1)

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Thanks! I put that in to see if anyone actually read this article.

Actually, it’s quite dangerous to have octonions be \textbb{O} and integral octonions be \textbf{O}. One character makes a big difference! Hermann Nicolai and coauthors use \texttt{O} for integral octonions, but that font doesn’t seem to work here.

Posted by: John Baez on July 25, 2013 5:27 AM | Permalink | Reply to this

Re: Integral Octonions (Part 1)

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After Kapranov’s lecture on ‘super’ things arising from a 2-truncation of the full stable-sphere spectrum- $\infty$ -story, we had a little speculation scattered about the nForum, e.g., here and after, about the full picture.

There was an idea (which I can’t find now) that already the move to ‘super’ is giving you a great deal of what you want. For example, Urs finds a bigger version of the brane scan, once one moves to the super-world, called the ‘brane bouquet’.

Naturally, true to my tutelage from you, I’d like to press on to the untruncated picture, having imbibed the mantra ‘the sphere spectrum is the true integers’, qua free abelian $\infty$ -group on one generator.

So how does the ‘super’ of your ‘Lie n-superalgebras’ fit with the appearance of the normed division algebras? What would you have seen of them had you remained at the level of Lie n-algebras?

Is there anything resembling what happened with the branes? You can build them by hand, but they appear naturally when we move to the 2-truncated super level.

Posted by: David Corfield on July 25, 2013 1:46 PM | Permalink | Reply to this

Re: Integral Octonions (Part 1)

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Luckily Kapranov and I both visited Nottingham during the British Mathematical Colloquium in Sheffield this spring, and we rode the train back together, and he explained his ideas on what I would call superdupersymmetry, namely letting the stable fundamental 2-group of the sphere spectrum act on things. I think this is great. I’ll have to watch that lecture sometime—thanks.

So how does the ‘super’ of your “Lie n-superalgebras” fit with the appearance of the normed division algebras. What would you have seen of them had you remained at the level of Lie $n$ -algebras?

You mean, maybe, what interesting Lie $n$ -algebras could we construct using normed division algebras?

Of course all the compact simple Lie groups and their Lie algebras arise, one way or another, from the normed division algebras! The $SO(n)$ ’s arise from real Hilbert spaces, the $SU(n)$ ’s arise from complex Hilbert spaces, and the $Sp(n)$ ’s arise from quaternionic Hilbert spaces. This is one of Arnol’d’s ‘trinities’, which Freeman Dyson calls the ‘three-fold way’. I explored how all three cases fit together here.

Then we’re left with the 5 exceptional groups, which are all connected to the octonions. $\mathrm{G}_2$ is the automorphism group of the octonions, while $\mathrm{F}_4, \mathrm{E}_6, \mathrm{E}_7,$ and $\mathrm{E}_8$ are the isometry groups of the Rosenfeld projective planes over $\mathbb{R} \otimes \mathbb{O}, \mathbb{C} \otimes \mathbb{O}, \mathbb{H} \otimes \mathbb{O}$ and $\mathbb{O} \otimes \mathbb{O}$ . (These still hold mysteries.)

So, we can say that the normed division algebras have been so helpful in creating the Lie algebras that, on the one hand, it would be greedy to demand they continue to help us as much in creating interesting Lie $n$ -algebras… but on the other hand, we might easily expect them to keep on giving, like reliable wealthy donors to the local charities.

If we want to build a Lie $n$ -algebra extending a Lie algebra $\mathfrak{g}$ of a simple Lie group, we can start by looking at Lie algebra $(n+1)$ -cocycles on $\mathfrak{g}$ . These give the most elementary kind of Lie $n$ -algebras: those with $\mathfrak{g}$ in degree zero and nontrivial extra stuff only in degree $n$ . I guess you’re familiar with this idea, thanks to HDA6.

These cocycles are well-known and thoroughly worked out, and Urs and others have worked out the corresponding Lie $n$ -groups and some of their applications to geometry and physics. Are these cocycles connected to division algebras in an interesting way? I don’t see how, but I haven’t really thought about it. It would be fun to think about cocycles on exceptional Lie algebras this way.

What’s strikingly apparent is that $(n+1)$ -cocycles on some famous Lie superalgebras can be built using normed division algebras. The reason is that normed division algebras give rise to beautiful special interactions between vectors and spinors. Or, if you prefer, they arise from beautiful special interactions between vectors and spinors.

It all boils down to the fact that if $\mathbb{K}$ is a normed division algebra, the 2 × 2 hermitian matrices valued in $\mathbb{K}$ can be seen as vectors in Minkowski spacetime. $\mathbb{K}^2$ is the space of spinors. So vectors act on spinors simply by matrix multiplication.

What next?

Beyond these very elementary Lie $n$ -algebras, which are like layer cakes with only two layers, there are many more, with more layers, which can interact in more complicated ways. I don’t know if anyone has started ‘classifying’ these, or some manageable bunch of them.

There’s also another obvious idea, which is to start categorifying the division algebras themselves. You can build some associative $n$ -algebras from Hochschild $(n+1)$ -cocycles on associative algebras, again following the layer cake philosophy. I was going to try this for the associative normed division algebras, and also see if something worked in the nonassociative case, but then I got busy doing other things. So someone should try this and let me know what happens. I would like to see ‘2-octonions’, or $n$ -octonions for whatever value of $n$ is relevant. But I don’t understand cohomology for nonassociative algebras.

But right now it’s ‘octonionic arithmetic’ and its relation to number theory and supersymmetric cosmological billiards that has me enthralled.

Posted by: John Baez on July 26, 2013 4:16 AM | Permalink | Reply to this

Re: Integral Octonions (Part 1)

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Kapranov

explained his ideas on what I would call superdupersymmetry, namely letting the stable fundamental 2-group of the sphere spectrum act on things.

So one question is

What if we didn’t truncate? What happens if we let the sphere spectrum itself act on things?

Anyway,

But right now it’s ‘octonionic arithmetic’ and its relation to number theory and supersymmetric cosmological billiards that has me enthralled.

And I see the next Part has appeared, so I’m off there now.

Posted by: David Corfield on July 26, 2013 2:12 PM | Permalink | Reply to this

Re: Integral Octonions (Part 1)

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This will be a newbie question. But what is the relationship to reciprocity laws to this? Similar to how standard modular forms relate to reciprocity laws in number fields?

Posted by: Archisman Rudra on July 27, 2013 2:25 PM | Permalink | Reply to this

Re: Integral Octonions (Part 1)

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Alas, I don’t know the answer to that. I know a bit about standard modular forms and reciprocity laws, but not how they’re related. I know the Langlands conjectures involve more general modular forms, and are sometimes considered a dramatic generalization of reciprocity laws, so maybe a special case of them form part of the answer to your question. I’ll ask someone I know.

What I mainly know, so far, is that certain generalized modular forms called Maass forms are important in the applications of $PSL(2,\mathbb{Z})$ and $PSL(2,\mathbf{O})$ to ‘quantum cosmological billiards’ (see Part 2). This sounds erudite but actually it’s pretty obvious. The simplest sort of Maass form is an eigenfunction of the Laplacian on the hyperbolic plane that’s invariant under the action of $PSL(2,\mathbb{Z})$ . Since this Laplacian is the Hamiltonian for a quantum billiard ball moving around on the hyperbolic plane, and for cosmology we want functions invariant under $PSL(2,\mathbb{Z})$ , it’s easy to see how Maass forms could be important.

Then the same Maass form idea should apply to gravity in 11 dimensions, using $PSL(2,\mathbb{O})$ and its integral form $PSL(2,\mathbf{O})$ . This is something I think I may be able to understand in more detail someday.

But I will need a lot of help to make serious inroads into the number-theoretic aspects of $PSL(2,\mathbf{O})$ .

Posted by: John Baez on July 28, 2013 8:46 AM | Permalink | Reply to this

Read the post Integral Octonions (Part 3)
Weblog: The n-Category Café
Excerpt: Let's take a look at the E8 lattice and the Lie algebra by the same name.
Tracked: July 30, 2013 2:21 PM

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July 23, 2013