"I've got fireball protection." said Oz cockily "I'm invincible.". He looked at the Wizard haughtily. "Do your worst.".
"O, didn't I tell you?" asked the Wizard, raising his arm, "Fireball protection wears off after 6 1/2 days.".
Oz paused for a moment and then ran to his seat. He sat attentively, notebook open and pencil in hand, like a good little student.
"That's better." said the Wizard.
"Now that we've seen cups and caps, I want to show you why duals are rotations and not reflections. We know that, when we see the diagram
|
|
|
V ^ _______
| / \
| / \
| V v |
\ / |
\_______/ |
V ^
|
|
|
we want to pull it straight to get
|
|
V ^
|
|
So, what do you suppose we want to do with
|
|
|
| _______
| / \
| / \
W ^ V v |
| | |
| | |
| / \ |
| | T | |
| \_/ |
| | |
| | |
| W v V ^
\ / |
\_______/ |
|
|
|
|
where T: V -> W? Well, if you do a calculation
(which is your homework, BTW), you find that this equals
|
|
W ^
|
|
/ \
| T*|
\_/
|
|
V ^
|
|
But, if you physically pull the first diagram straight,
you'll be rotating T to get T*. So, duals are rotations.".
"Another way to see that duals are rotations" said Toby the Acolyte "is that (V (x) W)* is W* (x) V*, not V* (x) W*. That is, the dual of
| |
| |
V v W v
| |
| |
is
| |
| |
W ^ V ^
| |
| |
which you get by rotating, not reflecting.".
"Hey!" said the Wizard "You're not supposed to be drawing diagrams. Give me that wand!".
Toby the Acolyte tossed the wand he'd been using over to Wizard, leaving showering sparks of drawing particles in an epicylic parabolic trajectory.
"Yes," said the Wizard, as he caught the wand and put it in his bag of tricks, "the most natural definition of (V (x) W)* is W* (x) V*, not V* (x) W*. Of course, we know that these 2 spaces are really the same, but our diagrams don't, as I've mentioned from time to time.
But! -- today we're going to teach our diagrams to do this. There is a natural map from BV,W: V (x) W -> W (x) V called the "braiding". Why is it called that? Well, let me draw it:
| |
| |
V v v W
\ /
\ /
/
/ \
/ \
W v v V
| |
| |
As you can see, V goes underneath W, just like 2 strands of a braid.
And notice that we can only do this by using 3 dimensions in our diagrams.
The reason our diagrams couldn't do this before
is that they had only 2 dimensions.
Imagine you were a resident of Flatland. If V and W were 2 objects, you could allow them to switch places. You could move them around one another. That's because Flatlander's spacetime has 3 dimensions, just like our diagrams now have. But, if you were a resident of Lineland, 2 objects could never switch places. That's because Lineland's spacetime has only 2 dimensions, like our diagrams had until today. Of course, we're better than Flatlanders; we have 4 spacetime dimensions. Later on, we'll see what that gets us in the diagrams.
Now consider this diagram:
| |
| |
W v v V
\ /
\ /
\
/ \
/ \
V v v W
| |
| |
This is not BW,V! BW,V is
| |
| |
W v v V
\ /
\ /
/
/ \
/ \
V v v W
| |
| |
Notice the difference in which side crosses over which.
| |
| |
W v v V
\ /
\ /
\
/ \
/ \
V v v W
| |
| |
rather, is the inverse of BV,W, BV,W-1. Why?".
"Well," said Toby the Acolyte "if you put one of them on top of the other, then you can just pull them apart.".
"Right," said the Wizard, like so:
| | | |
| | | |
V v v W | |
\ / | |
\ / | |
/ | |
/ \ | |
/ \ | |
W v v V = V v v W
\ / | |
\ / | |
\ | |
/ \ | |
/ \ | |
V v v W | |
| | | |
| | | |
or:
| | | |
| | | |
W v v V | |
\ / | |
\ / | |
\ | |
/ \ | |
/ \ | |
V v v W = W v v V
\ / | |
\ / | |
/ | |
/ \ | |
/ \ | |
W v v V | |
| | | |
| | | |
So, this explains why we draw BV,W-1 the way we do.
You want the topology to do the calculation for you.
This maneuver appears in knot theory; it's the 2nd Reidemeister move. In knot theory, you often study a knot by projecting it down into a plane, with overcrossings and undercrossings. Then you can use the Reidemeister moves to change one projection of the knot into another of the same knot.
There are 3 Reidemeister moves in all. The 2nd involves 2 lines; the 3rd involves 3 lines. It is:
| | | | | |
| | | | | |
U v v V v W U v V v v W
\ / / \ \ /
\ / / \ \ /
/ / \ /
/ \ / \ / \
/ \ / \ / \
| / = / |
\ / \ / \ /
\ / \ / \ /
/ \ / /
/ \ \ / / \
/ \ \ / / \
W v v V v U W v V v v U
| | | | | |
| | | | | |
See, you just slide things around. It's always W on top of V on top of U.
The 1st Reidemeister move is more subtle; it looks like this:
|
____ |
/ \ v V
/ \ /
/ \ /
V ^ /
\ / \
\ / \
\____/ v V
|
|
The cup at the bottom is just eV, of course.
And the cap at the top is -- wait a minute!
That cap looks a bit funny to me ... .".
Then John the Acolyte spoke up. "The cap goes the wrong way." he said It's not iV; the only way to interpret it is as iV*.".
"Exactly right!" said the Wizard "And that means that I drew the diagram wrong. It should look like this:
|
____ |
/ \ v V
/ \ /
/ \ /
V ^ /
\ / \
\ / \
\____/ v V**
|
|
Then we get the cap iV* correctly.
Anyway, the 1st Reidemeister move says that this equals
|
|
V v
|
|
|
V** v
|
|
which is the natural inclusion of V into V**.
Of course, if you identify V with V**, this is just the identity.
Oh, by the way, notice the braiding in the middle -- that's BV**,V.
So, there's a theorem in knot theory: Any 2 pictures of the same knot (or, more generally, of the same tangle) can be linked by a finite sequence of these 3 Reidemeister moves, together with diffeomorphisms applied to the picture, which are the identity at the boundary."
"Diffeo-what?" asked Oz.
"Never mind." said the Wizard. "Just imagine that the picture is drawn on a square rubber sheet. In addition to the Reideimeister moves, we're allowed to warp this rubber sheet as much as we want, as long as we don't move the boundary around.
Now, in the 1st Reidemeister move, we saw a cap that superficially looked like
_______
/ \
/ \
V ^ v V
| |
| |
But that's backwards from the usual cap,
so we were forced to interpret it as
_______
/ \
/ \
V ^ v V**
| |
| |
However, it turns out that, with the help of the braiding,
there is a way to interpret the backwards cap,
without looking at any identification of V with V**.
You simply define it to be this:
_
/ \
/ \
V v ^ V
\ /
\ /
/
/ \
/ \
V ^ v V
| |
| |
Notice that braiding BV*,V in the middle.
You can also define a backwards cup
| |
| |
V v ^ V
\ /
\_______/
as
| |
| |
V v ^ V
\ /
\ /
/
/ \
/ \
V ^ v V
\ /
\_/
Another way to think about the backwards cap and cup
is that they are the ordinary cap and cup for V* --
which involve V** -- combined with isomorphisms between V and V**.
The 1st Reidemeister move links this way of looking at them
with the diagrammatic definitions I just wrote.
But I'm not going to get into that now --
in fact, in real life, I never even mentioned it!".
"Erm, Mr Wizard, Sir?" asked Oz timidly "May I venture a question?".
The Wizard sighed. "If you must.".
"Well, I can't help but notice that all these crossings -- those in the recent diagrams I mean -- seem as if -- and here I could be wrong -- they could go either way. I mean, why use BV,V* instead of BV*,V-1? If you see what I mean, that is. Sir.". Oz gritted his teeth in anticipation.
The Wizard stared at Oz for a tense moment. "That's a very good question." he said at last. "And the answer is quite simple. It doesn't matter. In our case at least, it doesn't matter for a very profound reason. The reason is that diagrams for complex vector spaces are not 3 dimensional at all -- they're 4 dimensional.
It can be difficult to visualise 4 dimensional objects, even though -- or perhaps because -- we live in 4 dimensions. But consider the crossings
\ / \ / \ / \ / / and \ / \ / \ / \ / \Now, if you take the strand that's on top in the left crossing and move it slightly in the 4th dimension, then you can move it underneath the strand on bottom without intersecting them -- because the 4th coordinate will be different. Then you can move it back in the 4th dimension to get the crossing on the right. Therefore, these crossings are the same in a 4D diagram!
We can verify that complex vector spaces have 4D diagrams by simply checking that BW,V = BV,W-1. And this is obviously true.
One thing we learn from this is that the category of complex vector spaces works very well for studying knot theory in 4 dimensions. But we also see that knot theory in 4 dimensions is utterly trivial, because every knot can be untied by passing strands through each other. So, in order to do interesting knot theory -- 3D knot theory -- we need a category with 3 dimensional but not 4 dimensional diagrams. For example: the category of representations of a quantum group. Luckily, we're not doing knot theory in this course -- I'm just pointing it out along the way.
OK, it's time to put the backwards cap and cup to good use! What do you think of this diagram:
_______
/ \
/ \
| v V
| |
| |
| / \
V ^ | T |
| \_/
| |
| |
| v V
\ /
\_______/
This has the backwards cup on the top
and a linear operator T: V -> V on the right side.
So, what is it?".
"It's a complex number." said Miguel the Acolyte.
"Quite right" said the Wizard "a trivial observation, but one that must be made before we move on. But *which* complex number is it?".
"It's the trace of T." said Toby the Acolyte.
"Quite right" said the Wizard "and you know how we know that? Well, since everything is linear, it must depend linearly on T, and the complex number which depends linearly on T is the trace of T. Oh, and you can also calculate it and prove it.
So, what about this diagram:".
_______
/ \
/ \
V ^ v V
\ /
\_______/
"That's just the dimension of V." said Toby the Acolyte.
"Hmm," said the Wizard "it's nice to see you getting everything right, Toby, but you've got to let other people say something once in a while. Yes, this is just the trace of the identity map on V, which is the dimension of V. So circles have to do with dimension, a fact that only became fully appreciated in the late '80s.
So, with that understanding, we can now evaluate all sorts of pictures, such as:
____________
/ \
/ ___ \
/ / _ \ \
/ | |_v v \
| \___/ \
| _ \
| / \ \
| | \ v
| \ \ /
| \ \_/
| \
| \
| |
| ______/
| /
| \______
\ \
\ \
\ /
\_____________/
"No," said Jay the Acolyte "it's (dim V)3.".
"It's nice to see Toby get something wrong for once." said the Wizard "Yes, when things are next to each other -- and these circles are next to each other, using the 2nd Reidemeister move 4 times -- you multiply them, just like the tensor product. That funny picture evaluates to (dim V)3.
And I think we'll stop there for today.".