## Quantum Symmetry

Recall that Coquereaux likes looking at generalised Dynkin diagrams. In a paper on SU(3) graphs there is the example of the $\varepsilon_{21}$ graph, which generalises the diagram for $E_8$. This graph has 24 vertices in three sets of eight. Edge orientation is omitted from the diagram. Triality arises from considering the vertex labels mod 3. Complex conjugation corresponds to the symmetry of the central horizontal axis. There is a norm associated to this graph, which takes the value $0.5(1 + \sqrt{2} + \sqrt{6})$. The adjacency matrix for such a graph encodes an associative algebra structure which is compatible with the fusion algebra in a suitable way.

Apparently Ocneanu worked out many details of these quantum symmetries, but his work is largely unpublished. Meanwhile at The Cafe there is an amusing discussion in a comment thread initiated by the probably not-so-innocent David Corfield.

## 5 Responses so far »

1. 1

### lieven said,

Dear Kea,
just to let you know you’re not alone in being slightly annoyed by recent events in the blogosphere.
for now, i refrain from posting as otherwise sentences similar to your last one might slip in and often such sentences have the habit of boomeranging back…
atb :: lieven.

2. 2

### kneemo said,

I found their 1998 matrix algebra paper to be a nice read. In particular read their SM related comments at the end of page 11.

3. 3

### nosy snoopy said,

Bonjourrrr 🙂

So, Kea, you are thinking of explaining what is speed of light, why it is the way it is, heh?

I’m little shy to ask you, just have found a paper (http://arxiv.org/abs/hep-th/0609022), they say string theory is non-linear science :-O, I wonder how does n-category theory connect to tensors? (And there were also dessin d’enfant pictures it seems? :-S)

4. 4

### Kea said,

Thank you for visiting, Lieven. I am so glad that somebody understands how I feel. But you are right, one should not let such comments slip in, and I apologise.

5. 5

### Kea said,

nosy snoopy, you are a gem. That looks like an interesting paper. And thanks, kneemo.