How to show that Yang-Baxter equation is the same as braid equation? The quantum Yang-Baxter equation is $R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}$. The braid equation is $R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23}$. It is said that these two equations are equivalent. How to prove this? Thank you very much.

Let $V$ be a vector space and let $\tau:V\otimes V\to V\otimes V$ to be the _flip_ transformation defined by $v\otimes w\mapsto w\otimes v$ for all $v,w\in V$.

I believe what is meant by saying that the quantum Yang-Baxter and braid equations are "equivalent" is that if $R:V\otimes V\to V\otimes V$ is an invertible linear transformation, then $R$ satisfies one of them if and only if $\tau R$ satisfies the other. In other words, once you have a solution to one, you automatically get a solution to the other using the flip.