# Pictorial explanation of Dynkin index and quadratic Casimir?

Draw a colored loop. You explained quantum dimension. :-) OK, now you may use trivalent nodes and knot crossings too (with edges colored by irreps). You can now pictorialize 6j symbols, writhe normalizers, structure constants etc. My question is whether other things that also pop up all the time in Lie groups, as the (quadratic) Casimir or Dynkin index of an irrep can be visualized this way. (Since that's the only way I can do math :-)

Background: a) In "Birdtracks" an additive formula for Dynkin indices (multiplied with dimensions somehow) of the irreps involved in a Clebsch-Gordan expansion is given, b) I deduced another formula involving writhe normalizers and dimensions that is also valid for any Clebsch-Gordan expansion. (In short: Let $R_i\bigotimes{R_j}=\bigoplus_k{R_k}$, and $W_k=w_i*w_j/w_k$, where the $w_k$ are writhe normalizers, let $o_k$ be the dimensions, then $\Sigma_k{(W_k-W_k^{-1})o_k}=0$. Using pictures the proof is an one-liner.) It's natural to speculate whether a) and b) are essentially the same (or maybe even more things are additive over a Clebsch-Gordan expansion).

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In a), are you referring to (7.24) in "Birdtracks"? –  Bruce Westbury Jul 16 '12 at 12:39
Could you give an example of b)? –  Bruce Westbury Jul 16 '12 at 12:40
Is there any reason to think that there is a relationship other than the observation that both are sums over the composition factors in a tensor product? –  Bruce Westbury Jul 16 '12 at 12:41
@ Bruce Westbury: a) I don't know how many different versions of the ebook float around. I referred to the first table (8.1) in §18 about the E6 family. I assumed (by instinct) for the other families (I worked with E7) similar equations exist. –  Hauke Reddmann Jul 17 '12 at 12:55
@ Bruce Westbury: Regarding your two other questions, I proved that equation "pictorially". I try my best to sketch it in ASCII art: I write >- for a trivalent node and % for a knot crossing. The first one is colored with 3 colors, the latter with 2. %>- = Wijk * >- (structure constants). (By using Reidemeister 3 for nodes, Wijk=wk^2/Wjik.) By Schur, -<>= vanishes (if - and = are different colors). My gauge is setting all theta graphs to 1 which means that if - and = are the same color this graph reduces to - divided by the dimension of the color -, independent of the colors of <>. (contd.) –  Hauke Reddmann Jul 17 '12 at 13:11

## 1 Answer

It really sounds like you'd want to look at works of Pierre Vogel, starting from The universal Lie algebra, and also check recent results that use the same setup, like this one.

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I already mentally marked the work on the universal Lie algebra as very relevant. Unfortunately, the (recent) example you gave is exactly the stuff I can't understand due to having no formal education in higher math. (That's why I'm so desperately asking for pictures...) I mean, knot theory (on my level) can be summed up, so to say, in the three Reidemeister moves. Now if anyone could do the same for Lie algebras? ;-) –  Hauke Reddmann Jul 17 '12 at 12:47
To be honest, I strongly believe that the beauty of this story would motivate one to learn the necessary background maths rather than search for some mechanic manipulations not getting closer to actually understanding the topic. (I might sound a bit harsh now, but that's what your comment sounds like to me.) –  Vladimir Dotsenko Jul 17 '12 at 14:08