## Ribbon categories

In the last post I discussed the category of framed oriented tangles, which according to Shum’s theorem is a free ribbon category. As a corollary to Shum’s theorem, we may derive tangle invariants from any ribbon category. Let’s see how this works for the Kauffman bracket.

Consider planar diagrams, that is curves in the plane. These are like tangle diagrams only without self-intersections, i.e. no crossings. Just like tangles, they form a monoidal category since we can place them side by side or atop each other. Also just like tangles they have duality cups and caps.

Cup and Cap

Inspired by the definition of the Kauffman bracket, we extend the category of planar diagrams by linear combinations with coefficients polynomials in $A,A^{-1}$ and mod out by the circle relation:

Circle Relation

This gives a braiding and twist as in the calculations for the Kauffman bracket.

Braiding

Twist

The resulting ribbon category is called the Temperley-Lieb category, named for mathematicians who studied its implications in the context of statistical mechanics.

Now we have two examples of ribbon categories, the category of tangles and the Temperley-Lieb category. How else can we generate examples of ribbon categories? Recall that the category of finite dimensional vector spaces and linear maps formed a monoidal category with duals. We consider the subcategory of representations of an algebra $A$.

An algebra is a vector space in which we have a multiplication and unit with the familiar properties of associativity and unitality. For example, given a vector space $V$ the space $End(V)$ of endomorphisms of $V$, that is linear maps $V\to V$, forms an algebra where our multiplication is composition of linear maps and our unit is the identity map $1_V$. $V$ is a representation of $A$ iff there is a linear map $\rho_V:A\to End(V)$ which preserves multiplication and unit.

A Hopf algebra, in addition to having a multiplication and unit, also has maps $\Delta:A\to A\otimes A, \eta:A\to k$ called comultiplication, counit which are coassociative, and counital where $k$ is the field of scalars. This guarantees that the category $Rep_{fd}(A)$ of finite dimensional representations of $A$ is monoidal since we can define representations $\rho_{V\otimes W}=(\rho_V\otimes \rho_W)\Delta$ and $\rho_k=\eta$. We also require a map $S:A\to A$ called the antipode which switches the order of multiplication and is the convolution inverse to the identity. This guarantees that $Rep_{fd}(A)$ has left duals with $\rho_{V^*}=\rho_V S$.

If there are elements $R\in A\otimes A,h\in A$ such that $P_{V,W}(\rho_V\otimes\rho_W)(R)$ is a braiding, where $P_{V,W}:V\otimes W\to W\otimes V$ is the swap map $P_{V,W}(v\otimes w)=w\otimes v$ and where $\rho_V(h)$ is a twist, then we call $A$ a ribbon Hopf algebra. Clearly then $Rep_{fd}(A)$ is a ribbon category.

Surprisingly, ribbon Hopf algebras turn up in the study of Lie algebras. One may “quantize” a Lie algebra, deforming it by a formal parameter meant to mimic Plank’s constant $\hbar$ and the result is a ribbon Hopf algebra. This discovery led to a whole slew of new invariants and a new understanding of old invariants. For instance the Jones’ polynomial and the Kauffman bracket are related to the quantization of the most basic Lie algebra $sl(2,\mathbb{C})=su(2)\otimes\mathbb{C}=so(3)\otimes\mathbb{C}$. Invariants of tangles derived from quantized Lie algebras are called Reshetikhin-Turaev invariants or simply quantum invariants. When applied to links they give polynomials in a variable $q=e^\hbar$.

### 5 Responses to “Ribbon categories”

1. Jakob Blaavand Says:

Hi

I am about to start writing a note on the Jones Polynomial which shall be used at a summer school this comming July.

The only problem with knot theory and LaTeX is all the drawings. I see that you have a lot of the diagrams I need for this note. So I would like to ask you, how have you made these?

Regards
Jakob Blaavand

2. Eitan Says:

Hi Jakob, I use the xy-pic latex package to generate knot diagrams. Here’s some links that can help you to use xy-pic. Good luck with your project!

3. Jakob Blaavand Says:

Thanks a lot!

Regards
Jakob