# October 6, 1996 {#week91}

For a while now I've been meaning to finish talking about monads and
adjunctions, and explain what that has to do with the 4-color theorem.
But first I want to say a little bit more about "triality", which was
the subject of ["Week 90"](#week90).

Triality is a cool symmetry of the infinitesimal rotations in
8-dimensional space. It was only last night, however, that I figured out
what triality has to do with 3 dimensions. Since it's all about the
number *three* obviously triality should originate in the symmetries of
*three*-dimensional space, right? Well, maybe it's not so obvious, but
it does. Here's how.

Take good old three-dimensional Euclidean space with its usual basis of
unit vectors $i$, $j$, and $k$. Look at the group of all permutations of
$\{i,j,k\}$. This is a little 6-element group which people usually call
$S_3$, the "symmetric group on 3 letters".

Every permutation of $\{i,j,k\}$ defines a linear transformation of
three-dimensional Euclidean space in an obvious way. For example the
permutation $p$ with
$$p(i) = j, \quad p(j) = k, \quad p(k) = i$$
determines a linear transformation, which we'll also call $p$, with
$$p(ai+ bj + ck) = aj + bk + ci.$$
In general, the linear transformations we get this way either preserve
the cross product, or switch its sign. If $p$ is an even permutation
we'll get
$$p(v)\times p(w) = p(v\times w)$$
while if $p$ is odd we'll get
$$p(v)\times p(w) = -p(v\times w) = p(w\times v).$$
That's where triality comes from. But now let's see what it has to do
with *four*-dimensional space. We can describe four-dimensional space
using the quaternions. A typical quaternion is something like
$$a + bi + cj + dk$$
where $a$, $b$, $c$, $d$ are real numbers, and you multiply quaternions by using
the usual rules together with the rules
$$
  \begin{gathered}
    i^2 = j^2 = k^2 = -1
  \\ij=k,\quad jk=i,\quad ki=j,
  \\ji=-k,\quad kj=-i,\quad ik=-j.
  \end{gathered}
$$
Now, any permutation $p$ of $\{i,j,k\}$ also determines a linear
transformation of the quaternions, which we'll also call $p$. For
example, the permutation $p$ I gave above has
$$p(a + bi + cj + dk) = a + bj + ck + di.$$
The quaternion product is related to the vector cross product, and so
one can check that for any quaternions $q$ and $q'$ we get
$$p(qq') = p(q)p(q')$$
if $p$ is even, and
$$p(q'q) = p(q')p(q)$$
if $p$ is odd. So we are getting triality to act as some sort of
symmetries of the quaternions.

Now sitting inside the quaternions there is a nice lattice called the
"Hurwitz integral quaternions". It consists of the quaternions
$a + bi + cj + dk$ for which either $a$, $b$, $c$, $d$ are all integers, or all
half-integers. Here I'm using physics jargon, and referring to any
number that's an integer plus $1/2$ as a "half-integer". A half-integer
is *not* any number that's half an integer!

You can think of this lattice as the $4$-dimensional version of all the
black squares on a checkerboard. One neat thing is that if you multiply
any two guys in this lattice you get another guy in this lattice, so we
have a "subring" of the quaternions. Another neat thing is that if you
apply any permutation of $\{i,j,k\}$ to a guy in this lattice, you get
another guy in this lattice --- this is easy to see. So we are getting
triality to act as some sort of symmetries of this lattice. And *that*
is what people *usually* call triality.

Let me explain, but now let me use a lot of jargon. (Having shown it's
all very simple, I now want to relate it to the complicated stuff people
usually talk about. Skip this if you don't like jargon.) We saw how to
get $S_3$ to act as automorphisms and antiautomorphisms of $\mathbb{R}^3$ with its
usual vector cross product... or alternatively, as automorphisms and
antiautomorphisms of the Lie algebra $\mathfrak{so}(3)$. From that we got an action
as automorphisms and antiautomorphisms of the quaternions and the
Hurwitz integral quaternions. But the Hurwitz integral quaternions are
just a differently coordinatized version of the $4$-dimensional lattice
$D_4$! So we have gotten triality to act as symmetries of the $D_4$
lattice, and hence as automorphisms of the Lie algebra $D_4$, or in other
words $\mathfrak{so}(8)$, the Lie algebra of infinitesimal rotations in 8 dimensions.
(For more on the $D_4$ lattice see ["Week 65"](#week65), where I
describe it using different, more traditional coordinates.)

Actually I didn't invent all this stuff, I sort of dug it out of the
literature, in particular:

1) John H. Conway and Neil J. A. Sloane, _Sphere Packings, Lattices and Groups_, second edition, Grundlehren der mathematischen Wissenschaften **290**, Springer-Verlag, 1993.

and

2) Frank D. (Tony) Smith, "Sets and $C^n$; quivers and $A$-$D$-$E$; triality; generalized supersymmetry; and $D_4$-$D_5$-$\mathrm{E}_6$", preprint available as [`hep-th/9306011`](https://arxiv.org/abs/hep-th/9306011).

But I've never quite seen anyone come right out and admit that triality
arises from the permutations of the unit vectors $i$, $j$, and $k$ in 3d
Euclidean space.

I should add that Tony Smith has a bunch of far-out stuff about
quaternions, octonions, Clifford algebras, triality, the $D_4$ lattice ---
you name it! --- on his home page:

3) Tony Smith's home page, `http://valdostamuseum.org/hamsmith/`

He engages in more free association than is normally deemed proper in
scientific literature --- you may raise your eyebrows at sentences like
"the Tarot shows the Lie algebra structure of the $D_4$-$D_5$-$\mathrm{E}_6$ model, while
the I Ching shows its Clifford algebra structure" --- but don't be
fooled; his mathematics is solid. When it comes to the physics, I'm not
sure I buy his theory of everything, but that's not unusual: I don't
think I buy *anyone's* theory of everything!

Let me wrap up by passing on something he told me about triality and the
exceptional groups. In ["Week 90"](#week90) I described how you
could get the Lie groups $\mathrm{G}_2$, $\mathrm{F}_4$ and $\mathrm{E}_8$ from triality. I didn't know how
$\mathrm{E}_6$ and $\mathrm{E}_7$ fit into the picture. He emailed me, saying:

> "Here is a nice way: Start with $D_4 = \mathrm{Spin}(8)$:
>
> $$28 =  28  +   0  +   0  +   0  +   0  +   0  +   0$$
>
> Add spinors and vector to get $\mathrm{F}_4$:
>
> $$52 =  28  +   8  +   8  +   8  +   0  +   0  +   0$$
>
> Now, "complexify" the $8+8+8$ part of $\mathrm{F}_4$ to get $\mathrm{E}_6$:
>
> $$78 =  28  +  16  +  16  +  16  +   1  +   0  +   1$$
>
> Then, "quaternionify" the $8+8+8$ part of $\mathrm{F}_4$ to get $\mathrm{E}_7$:
>
> $$133 =  28  +  32  +  32  +  32  +   3  +   3  +   3$$
>
> Finally, "octonionify" the $8+8+8$ part of $\mathrm{F}_4$ to get $\mathrm{E}_8$:
>
> $$248 =  28  +  64  +  64  +  64  +   7  +  14  +   7$$
>
> This way shows you that the "second" $\mathrm{Spin}(8)$ in $\mathrm{E}_8$ breaks down as
> $28 = 7 + 14 + 7$ which is globally like two 7-spheres and a $\mathrm{G}_2$, one $S_7$ for
> left-action, one for right-action, and a $\mathrm{G}_2$ automorphism group of
> octonions that is needed to for "compatibility" of the two $S_7$s. The
> $3+3+3$ of $\mathrm{E}_7$, the $1+0+1$ of $\mathrm{E}_6$, and the $0+0+0$ of $\mathrm{F}_4$ and $D_4$ are the
> quaternionic, complex, and real analogues of the $7+14+7$."

When I asked him where he got this, he said he cooked it up himself
using the construction of $\mathrm{E}_8$ that I learned from Kostant together with
the Freudenthal-Tits magic square. He gave some references for the
latter:

4) Hans Freudenthal, _Adv. Math._ **1** (1964) 143.

5) Jacques Tits, _Indag. Math._ **28** (1966) 223--237.

6) Kevin McCrimmon, "Jordan Algebras and their applications", _Bull. AMS_ **84** (1978) 612--627, at pp. 620-621. Available at [`http://projecteuclid.org`](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183540925)

I would describe it here, but I'm running out of steam, and it's easy
to learn about it from his web page:

7) Tony Smith, Freudenthal-Tits magic square, `http://valdostamuseum.org/hamsmith/FTsquare.html`

------------------------------------------------------------------------

> *"I regret that it has been necessary for me in this lecture to
administer such a large dose of four-dimensional geometry. I do not
apologise, because I am not really responsible for the fact that nature
in its most fundamental aspect is four-dimensional"*
>
> --- Albert North Whitehead.