Intro

People usually ask you what math is good for. As simple as that. Best is to have a good answer here, preferably involving quantum groups.

To start with, you need a pack of cigarettes, for instance Marlboro Red. Have one handy, whenever you go out, and at home too:

marlboro

Now let’s put this pack on the table, and rotate it. Not many symmetries! However, in 3D you obviously have more symmetries. So conclusion:

The more the dimensions, the more the symmetries.

In other words, we regular 3D folks are definitely smarter than 2D folks. This is certainly the case, no wonder we’re such an advanced civilization:

gargamel

However, if you think a bit more at all this, it’s truly scary. Imagine that the pack of Marlboro has even more symmetries in 4D, or more!

That would be certainly interesting, scientifically speaking, and anyone knowing such things could develop freakish new technologies:

mars_attacks

Quite surprisingly, the pack of Marlboro does have an infinity of symmetries, in enough dimensions, say in infinite dimensions.

These “alien” symmetries are, fortunately, known to humans. At least since 20 years or so. And they form a so-called quantum symmetry group.

All this is related to the well-known fact that the elementary particles live in an infinity of dimensions too. So, quantum groups are expected to be useful in connection with various quantum mechanical phenomena:

Trinity_Test_Fireball_16ms

The structure of quantum groups is not known, but there is a lot of good work in this direction. Nor is the precise link with quantum mechanics. As for potential applications, we are still far away from them.

Finally, there is some poetry in this too. By some kind of miracle, the world at small scales is related, in a subtle way, to the world at big scales.

So, there should be as well a link of all this with astrophysics:

Crab_Nebula

That’s all folks, and more from us soon. Do not try at home.


If interested in all this, there’s some math to be learned:

1. Following Weyl and others, let’s look at the compact Lie groups. These are the usual groups of matrices G < U_N, which are closed. The group operation is of course something very simple, (UV)_{ij}=\sum_kU_{ik}V_{kj}.

2. Now if you want quantum groups, the matrix coordinates u_{ij}:U\to U_{ij} should no longer commute. You need here operator algebras.

3. You're led in this way into algebras of type A=C(G), with coordinates u_{ij}, such that there is a map \Delta(u_{ij})=\sum_ku_{ik}\otimes u_{kj}.

4. Here G is your quantum group. Of course this does not exist, as a concrete object. But no problem. After all, particles don't quite exist, either.

5. With this in hand, you can do many things. If you consider the algebra C(S_N^+) generated by the entries of a "magic unitary", whatever that means, you have here your quantum permutation group S_N^+. Nice.

6. Now given a cigarette pack, or whatever discrete object, you can talk about its quantum symmetry group, as subgroup of an appropriate S_N^+.


Still following? If this sounds exciting, and not too technical, you can check the intro of my quantum group book, for more advertisement.

And if you like that too, you’re good for reading the whole book. That’s 200 pages, graduate level. Takes a few weeks, and you’re in.

Alternatively, you can check this blog, and get to books afterwards.