My talk at "Advances in Quantum and Low-Dimensional Topology", University of Iowa, March 2016.

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Suppose you are interested in
the representation theory of

There's a trivial 1-dimensional representation.

There's an obvious 3-dimensional representation, called the "defining" representation.

There are morphisms between representations.

There are operations on morphisms:

- compose
- tensor product
- dual
- linear combinations

Kuperberg's A2 spider encodes all of this with trivalent "web" diagrams.

There are trivalent vertices with arrows either all-in or all-out. Roughly, a trivalent vertex is the cross product, and a cap is the dot product.

- compose = stack vertically
- tensor product = place side by side
- dual = reflect vertically

Linear combinations of morphisms are just formal linear combinations of diagrams. Composition and tensor product are multilinear. Taking the dual is conjugate linear.

There are relations that let you simplify
bubbles, digons and squares.
There's a unique consistent way to
sprinkle $q$s into the defining relations.
See here.
So the spider for

I won't define $U_q\mathfrak{sl}_3$ here. Whatever it is, you can study its representation theory by drawing doodles!

Here is a new way to encode the representation theory of $U_q\mathfrak{sl}_3$ by drawing doodles.

Start with the root system for SL(3), and the weights for the defining representation:

Make a choice of positive roots: red, green and blue.

A "diagram" will be a triple of oriented Temperley-Lieb diagrams, one red, one green, and one blue, superimposed. Strands of different colors can pass through each other.

Instead of Kuperberg's up arrow, we use a sum of 3 pairs of arrows.

↑ ↑ + ↑ ↓ + ↓ ↓

Each term corresponds to a weight.
Each color corresponds to a root.
Take dot products

Instead of Kuperberg's vertex with arrows all-in, we have a sum of 6 terms, one for each of the 6 = 3! orders of the 3 weights.

If the 3 different pairs of edges go in to a vertex, there is a unique way to join them so the colors and orientations match.

Similarly for an all-out vertex.

Tensor = side by side. Dual = reflect vertically.

Compose by stacking. If the colors don't match you get zero.

Same for green and blue.

Our definition is isomorphic to Kuperberg's spider.

**Proof:**
Just check it satisfies Kuperberg's relations.

- Better proof: put $U_q\mathfrak{sl}_3$ and its representation theory in the same picture, like $U_q\mathfrak{sl}_2$.
- Rediscover Ocneanu cells, as in Evans and Pugh.
- Rediscover a categorification, as in Morrison and Nieh, using a colored oriented version of Dror's treatment of Khovanov homology.
- Do all of this for other Lie algebras.

I should have mentioned crossings. In my picture they have a fairly natural definition where each color does its own Kauffman bracket.

I don't know what happens if $q$ is an $n$th root of unity. Maybe allow $n$-valent vertices with arrows all-in or all-out?