# Braids and the Yang-Baxter equation

I recently gave a talk on the Yang-Baxter equation. The focus of the talk was to state the connection between the Yang-Baxter equation and the braid relation. This connection comes from a system of interacting particles. In this post, I’ll go over part of my talk. You can access the full set of notes here.

### Interacting particles

Imagine two particles on a line, each with a state that can be any element of a set $\mathcal{X}$. Suppose that the only way particles can change their states is by interacting with each other. An interaction occurs when two particles pass by each other. We could define a function $F : \mathcal{X} \times \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ that describes how the states change after interaction. Specifically, if the first particle is in state $x$ and the second particle is in state $y$, then their states after interacting will be $F(x,y) = (F_a(x,y), F_b(x,y)) = (\text{new state of particle 1}, \text{new state of particle 2}),$

where $F_a,F_b : \mathcal{X} \times \mathcal{X} \to \mathcal{X}$ are the components of $F$. Recall that the particles move past each other when they interact. Thus, to keep track of the whole system we need an element of $\mathcal{X} \times \mathcal{X}$ to keep track of the states and a permutation $\sigma \in S_2$ to keep track of the positions.

#### Three particles

Now suppose that we have $3$ particles labelled $1,2,3$. As before, each particle has a state in $\mathcal{X}$. We can thus keep track of the state of each particle with an element of $\mathcal{X}^3$. The particles also have a position which is described by a permutation $\sigma \in S_3$. The order the entries of $(x,y,z) \in \mathcal{X}^3$ corresponds to the labels of the particles not their positions. A possible configuration is shown below: A possible configuration of the three particles. The above configuration is escribed as having states and positions .

As before, the particles can interact with each other. However, we’ll now add the restriction that the particles can only interact two at a time and interacting particles must have adjacent positions. When two particles interact, they swap positions and their states change according to $F$. The state and position of the remaining particle is unchanged. For example, in the above picture we could interact particles $1$ and $3$. This will produce the below configuration: The new configuration after interacting particles and in the first diagram. The configuration is now described by the states and the permutation .

To keep track of how the states of the particles change over time we will introduce three functions from $\mathcal{X}^3$ to $\mathcal{X}^3$. These functions are $F_{12},F_{13},F_{23}$. The function $F_{ij}$ is given by applying $F$ to the $i,j$ coordinates of $(x,y,z)$ and acting by the identity on the remaining coordinate. In symbols, $F_{12}(x,y,z) = (F_a(x,y), F_b(x,y), z),$ $F_{13}(x,y,z) = (F_a(x,z), y, F_b(x,z)),$ $F_{23}(x,y,z) = (x, F_a(y,z), F_b(y,z)).$

The function $F_{ij}$ exactly describes how the states of the three particles change when particles $i$ and $j$ interact. Now suppose that three particles begin in position $123$ and states $(x,y,z)$. We cannot directly interact particles $1$ and $3$ since they are not adjacent. We have to pass first pass one of the particles through particle $2$. This means that there are two ways we can interact particles $1$ and $3$. These are illustrated below. The two ways of passing through particle 2 to interact particles 2 and 3.

In the top chain of interactions, we first interact particles $2$ and $3$. In this chain of interactions, the states evolve as follows: $(x,y,z) \to F_{23}(x,y,z) \to F_{13}(F_{23}(x,y,z)) \to F_{12}(F_{13}(F_{23}(x,y,z))).$

In the bottom chain of interactions, we first interact particles $1$ and $2$. On this chain, the states evolve in a different way: $(x,y,z) \to F_{12}(x,y,z) \to F_{13}(F_{12}(x,y,z)) \to F_{23}(F_{13}(F_{12}(x,y,z))).$

Note that after both of these chains of interactions the particles are in position $321$. The function $F$ is said to solve the Yang–Baxter equation if two chains of interactions also result in the same states.

Definition: A function $F : \mathcal{X} \times \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is a solution to the set theoretic Yang–Baxter equation if, $F_{12}\circ F_{13} \circ F_{23} = F_{23} \circ F_{13} \circ F_{12}.$

This equation can be visualized as the “braid relation” shown below. Here the strings represent the three particles and interacting two particles corresponds to crossing one string over the other.

• The identity on $\mathcal{X} \times \mathcal{X}$.
• The swap map, $(x,y) \mapsto (y,x)$.
• If $f,g : \mathcal{X} \to \mathcal{X}$ commute, then the function $(x,y) \to (f(x), g(y))$ is a solution the Yang-Baxter equation.