The n-Category Café

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.

Yay! Someone is actually asking questions about my lectures!

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!

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

on the bottom and then

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.

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).

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.

Wow! so the blog accepts animations!!

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!

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