## October 7, 2008

### Lie Theory Through Examples 2

#### Posted by John Baez

In this week’s seminar on Lie Theory Through Examples, we’ll move on up to the $A_3$ lattice.

You’ve seen this lattice before: when you stack spheres in a triangular pyramid, their centers lie at points that form a lattice of this sort:

But to derive this lattice starting with the Lie group $SU(4)$, we’ll need to talk a bit about the Killing form. That’s what allows us to measure angles and distances in the Lie algebra.

• Lecture 2 (Oct. 6) - The Killing form and the $A_3$ lattice.

A different view makes it clear why the A3 lattice is also called a "cubic close packing":

You can also think of the A3 lattice as built from octahedra and tetrahedra. This figure is from Buckminster Fuller’s patent for the "octet truss", now widely used in architecture:

In the $A_3$ lattice, each point has 12 nearest neighbors. These form the vertices of a cuboctahedron:

Posted at October 7, 2008 12:15 AM UTC

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### Re: Lie Theory Through Examples 2

Wait, was the lecture really on Monday?

Posted by: Toby Bartels on October 7, 2008 11:58 AM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

No, the lecture is today, Tuesday. Monday is my day for writing the lecture notes.

By the way, Lisa and I are going to hear the Obama–McCain debate which starts at 6:00 today, so I won’t be able to talk about smooth anafunctors after the seminar this time.

Posted by: John Baez on October 7, 2008 5:38 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

[…] talk about smooth anafunctors […]

Behind the scenes we have recently made some progress, it seems, with understanding smooth $\omega$-anafunctors (i.e. the ana-version of strict smooth $\infty$-functors on strict smooth $\infty$-categories).

If you, Toby, are interested, I’d be happy to share our notes on this and maybe talk /work about it together. (But not here in public for the time being.) Let me know.

Posted by: Urs Schreiber on October 7, 2008 6:31 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

By the way, Lisa and I are going to hear the Obama–McCain debate which starts at 6:00 today

Hey, that’s cool. I will be looking out for you when the camera pans through the crowd for questions. You should ask them a doozer!

Posted by: Bruce Bartlett on October 7, 2008 8:20 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

I didn’t mean we’d be physically present at the debate in Nashville Tennessee. We’ll either watch it on TV in the gym (pedaling helps relieve frustration) or at the big Obama party in downtown Riverside.

Posted by: John Baez on October 7, 2008 8:41 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Very interesting lecture. I got a bit confused at the end: I didn’t understand the part about the $A_3$ lattice and a “square pyramid”.

I’m thinking: What is a square pyramid? Is it a pyramid of oranges which has been rotated somehow? What do you mean at the top of the blog entry above when you say “A different view makes it clear why the $A_3$ lattice is also called a ‘cubic close packing’”? I know I’m supposed to just “see it” from this picture, but I don’t. It just looks like a bunch of balls in a cube. What should I be noticing?

Posted by: Bruce Bartlett on October 9, 2008 9:25 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Bruce wrote:

Very interesting lecture. I got a bit confused at the end…

Indeed, the end required some feats of visualization that are quite difficult without pictures, or even with pictures. But they’re worthwhile, because they reveal how cool the $A_3$ lattice really is.

I didn’t understand the part about the $A_3$ lattice and a “square pyramid”. I’m thinking: What is a square pyramid?

It’s the usual way people stack cannonballs:

Lay out an $n \times n$ square of balls, then lay out an $(n-1) \times (n-1)$ square on top of that, and so on. The result is an $A_3$ lattice!

What makes this interesting is that you also get an $A_3$ lattice from a pyramid of balls with an equilateral triangle as base! When you can see this is true, you really understand the $A_3$ lattice.

“A different view makes it clear why the A 3 lattice is also called a ‘cubic close packing’”? I know I’m supposed to just “see it” from this picture, but I don’t. It just looks like a bunch of balls in a cube. What should I be noticing?

First, you see right off that we could pack all of 3d space with cubes like this, and get a way of packing 3d space with spheres that has cubical symmetry. Lots of cubical crystals actually have the atoms packed in this pattern.

Second, you should see that the lattice underlying this picture is the same as the one in the square pyramid. We’ve got a layers of spheres arranged in a $n \times n$ square, then another layer arranged in an $(n-1) \times (n-1)$ square… but unlike the pyramid, we then put on a layer of size $n \times n$, and keep alternating like that. But, it’s still the $A_3$ lattice.

I explain in the notes why the $A_3$ lattice

$\{(a,b,c) : a,b,c \in \mathbb{Z}, a+b+c \; even \}$

has horizontal layers that consist of dots arranged in a square pattern. For example, this layer:

$\{(a,b,0) : a,b \in \mathbb{Z}, a+b \; even \}$

and then this layer

$\{(a,b,1) : a,b \in \mathbb{Z}, a+b odd \; \}$

and so on. The layers in the square pyramid or the cubic close packing above are just portions of this infinite lattice!

Posted by: John Baez on October 10, 2008 6:20 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Now I’m confused. In your picture of multi-coloured balls should they be of unit radius? No, wait! I see what the confusion is, it’s twisted 45 degrees about the $z$-axis from what I might expect. I was expecting, for instance, the red, orange and pink balls along the $x$-axis to be touching. I was thinking of having

(1)$\{(a,b,0)\}:a,b \in \mathbb{Z}, a and b even\}$

on the bottom and then

(2)$\{(a,b,\sqrt{2})\}:a,b \in \mathbb{Z}, a and b odd\}$

etc. But that’s just your lattice scaled by $\sqrt{2}$ and rotated 45 degrees about the $z$-axis. I was thinking about packing unit spheres rather than the naturality of the lattice, I guess.

Posted by: Simon Willerton on October 10, 2008 7:00 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

I have a question or a little request.

Are the words roots,weights and Cartan subalgebra going to come up in the lectures? Can they? I just need some little more information on those to make the connection between your lecture notes and Fulton and Harris.

thank you

Posted by: Christian on October 11, 2008 10:30 AM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Christian wrote:

Are the words roots,weights and Cartan subalgebra going to come up in the lectures?

Sure! I just happen to dislike the traditional way of teaching the theory of simple Lie algebras, so I’m approaching it from a different angle, more in line with J. F. Adam’s marvelous Lectures on Lie Groups. Lie algebras are just a technical tool for studying Lie groups, so it’s wiser to introduce all the concepts you mention starting from a Lie group perspective.

For various reasons, I prefer to start by studying compact simple Lie groups (denoted $K$) rather than complex ones (denoted $G$). For example, $K = SU(n)$ rather than $G = SL(n,\mathbb{C})$. There’s a one-to-one correspondence, so it’s just a different viewpoint. The Lie algebra of $K$ is a real Lie algebra. If we tensor it with the complex numbers we get the Lie algebra of $G$. There’s also a way to go back.

A Cartan subalgebra is the Lie algebra of a maximal abelian subgroup $A$ of $G$. I prefer to use the Lie algebra of a maximal abelian subgroup $T$ of $K$. Again, we get the Lie algebra of $A$ by tensoring the Lie algebra of $T$ with the complex numbers.

For example, if $G = SL(4,\mathbb{C})$, then $K = SU(4)$. $T$ is a 3d torus so its Lie algebra is a 3d real vector space. The Cartan subalgebra is a 3d complex vector space. It’s a lot easier to draw stuff in a 3d real vector space! All the pictures we’re drawing here live in this space.

Similarly, the other concepts you mention — roots and weights — are closely related to things I’m talking about, and they’ll eventually make their appearance in my notes.

For example, the weight lattice is the dual of the lattice I’m talking about (the kernel of the exponential map).

Posted by: John Baez on October 12, 2008 5:39 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Thanks so much for answering! This cleared up a lot.

And again, thanks for these beautiful lectures.

(I really can’t wait to get to E8! Will it be on November 4th when Obama gets elected? )

Posted by: Christian on October 12, 2008 10:09 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

I’m glad my remarks helped. I’ll say more later in the notes.

I sort of doubt I’ll get to $E_8$ by November 4th — but if it seems that Obama will win, I may talk about $E_8$ just to celebrate.

The odds look good now, but a lot can happen between now and then.

Posted by: John Baez on October 13, 2008 3:35 AM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Isn’t the $A_3$ lattice the same as the face-centred cubic (fcc) lattice? This isn’t completely obvious from your diagram (too many balls, I think) but it’s well known in crystallography, unless I’m much mistaken.

Posted by: Tim Silverman on October 11, 2008 5:15 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Yes, the “face-centered cubic” is yet another way of describing the $A_3$ lattice. Draw dots at the vertices of a cubical lattice and then draw an extra dot at the center of each face of each cube:

This isn’t completely obvious from your diagram (too many balls, I think)

I don’t think it’s just that. Different views of the $A_3$ lattice emphasize different aspects of this structure. While I consider myself pretty good at visualizing things, I find it quite a challenge to see how all these views are views of the same thing!

That’s what makes it so fun.

Posted by: John Baez on October 11, 2008 6:14 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

How do I “see” the “packing pennies” $A_2$ lattice in the face-centered cubic diagram above? On which plane can one pack pennies? I’m supposed to see a whole bunch of equilateral triangles, but somehow I just see right-angled ones. What’s going on?

Posted by: Bruce Bartlett on October 12, 2008 12:40 AM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Posted by: Greg Egan on October 12, 2008 2:01 AM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Greg Egan nails the “Proof Without Words” format of AMM!

Posted by: Jonathan Vos Post on October 12, 2008 7:33 AM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Thanks Greg.

Mmm, I’m wondering if there is any “God-given best” angle to look at the $A_3$ lattice. I find the triangular cannonball arrangement as on the top of page 6 of John’s notes quite intuitive, so could I make the following argument? I see from Wikipedia that Lead nitrate packs into an fcc pattern. Let’s ignore the nitrate ions. Imagine the following procedure: Stick a blob of lead nitrate in a room, then vaporize it, and then cool down the room and wait for it to form a crystal again. If we looked at it under a microscope, wouldn’t it be true that the whole thing would go layer by layer: it would start with the lead ions arranging themselves into a planar penny packing pattern on the floor, and then the next layer on top of that, and so on.

What they wouldn’t do is build up the face-centered-cubic pattern layer by layer, because the first layer (atoms at the corners of squares and one atom in the middle of each square) wouldn’t be an efficient planar pattern!

In other words I’m asking if it’s true or not that the triangular pyramid of cannonballs pattern is actually the natural way (from the point of solids) to look at the $A_3$ lattice, and not the face-centered-cubic pattern.

After all, the normal vector to the “floor” in our experimental lab gives us a natural direction to look at the crystal from. From that direction, it won’t look like a face-centered-cubic pattern.

Posted by: Bruce Bartlett on October 12, 2008 10:43 AM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

I can’t speak specifically to lead nitrate, but in general, no. The dynamics of crystal growth are complicated and subtle, they depend on conditions of temperature and pressure, and of the concentrations of the crystallising substances; atoms/ions/molecules may be quite mobile over newly-formed surfaces before bonding to their neighbours and locking into place.

Hence the immense variety of crystal forms found in nature, even for identical chemicals packed in identical lattices. Bear in mind, for example, the uniqueness of snowflakes.

(Cue: entrance of a real chemist who actually knows this stuff in detail … if any are lurking out there.)

Posted by: Tim Silverman on October 12, 2008 12:13 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Is there existing a 3D animation, ideally of the kind that lets you rotate the image on screen? cf. the associahedron

Posted by: jim stasheff on October 12, 2008 1:30 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

There is a nice website maintained by the Navy Research Lab that features a JAVA applet that allows you to rotate many crystal structures, including the fcc lattice.

It only shows one unit cell for the lattice as far as I can tell, but it’s a start. I like to rotate the cell so as to look down one of the threefold symmetry axes.

Posted by: Garett Leskowitz on October 16, 2008 8:03 AM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Cool! Nice to see you here, Garett!

To those who don’t know him, Garett Leskowitz is a chemist who studies stuff like magnetic resonance spectroscopy. He used to attend my seminar at UCR. Now he’s moved on to bigger and better things… but he’s getting into Lie theory, so he’s reading the seminar notes.

Posted by: John Baez on October 16, 2008 11:39 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Bruce wrote:

Mmm, I’m wondering if there is any “God-given best” angle to look at the $A_3$ lattice.

To me, it seems important to grok the $A_3$ lattice from numerous viewpoints, and gain the mental agility to switch easily between these viewpoints:

• Stack balls in a triangular pyramid. Their centers form an $A_3$ lattice.
• Stack balls in a square pyramid. Their centers form an $A_3$ lattice.
• Take a cubic lattice and draw a dot at each corner and the midpoint of each face. These dots form an $A_3$ lattice.
• Take a cubic lattice and draw a dot at the center of each cube and the midpoint of each edge. These dots form an $A_3$ lattice.
• Take a cubic lattice, color the cubes alternately red and black in a 3-dimensional analogue of a checkerboard pattern, and draw a dot at the center of each red cube. These dots form an $A_3$ lattice.

It’s taken me my entire life to realize that these are all different viewpoints on the same entity! Of course I wasn’t working on it full time, but I have bumped my head against all these descriptions many times.

Posted by: John Baez on October 12, 2008 6:41 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

It’s going to take me a good long time to “grok” all these viewpoints. I guess a journey of a thousand miles starts with a single step.

Posted by: Bruce Bartlett on October 13, 2008 10:30 AM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Bruce is asking about crystal growth of a face-centered cubic lattice (otherwise known as the $A_3$ lattice), while Jim wants to see 3d animations that you can rotate. Here’s something I found:

We can imagine this as the growth of a face-centered cubic crystal starting from a one-atom seed.

This animation comes from a crystallography website that’s pushing Buckminster Fuller’s viewpoint: down with cubes, up with the octet truss. Unfortunately the links to VRML files seem to be broken, including one that could be a rotatable face-centered cubic lattice. But here’s a nice picture that shows how the face-centered cubic lattice is also full of octahedra and cuboctahedra:

Posted by: John Baez on October 12, 2008 6:06 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Wow! so the blog accepts animations!!

Posted by: jim stasheff on October 13, 2008 1:25 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

These are just gif files — a commonly used format for pictures, which has the ability to play ‘loops’ of pictures. You’ll see lots of such short loops on Wikipedia.

I’ve never tried to include more complicated interactive animations here; these usually require javascript. I bet Jacques Distler could make this possible, but I’m not requesting it!

If you’ve never seen Greg Egan’s applets using javascript, go take a look!

Posted by: John Baez on October 13, 2008 5:19 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Theoretically, SVG supports animation, although I’ve never managed to get it to work. (Not that I need it, I was just curious.)

Posted by: Tim Silverman on October 13, 2008 8:15 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

Could you give you definition of dimension and minimum length as it used for this subject.

Posted by: jal on October 15, 2008 4:50 PM | Permalink | Reply to this

### Re: Lie Theory Through Examples 2

I’m using the usual definitions of ‘dimension’ and ‘length’ that apply to $n$-dimensional Euclidean space.

Posted by: John Baez on October 16, 2008 11:42 PM | Permalink | Reply to this

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