Yang–Baxter equation

Remarks

Let $\tau \colon V \otimes V \to V \otimes V$ be the flip $u \otimes v \mapsto v \otimes u$ and $c = \tau \circ R$. Then $R$ satisfies the Yang–Baxter equation if and only if $c$ satisfies the braid relation $c_{12} c_{23} c_{12} = c_{23} c_{12} c_{23}$, with $c_{12} = c \otimes \id_V$ and $c_{23} = \id_V \otimes c$. Example: $R = \tau$. On $u \otimes v \otimes w$, the left side sends it through $u \otimes w \otimes v$ and $v \otimes w \otimes u$ to $w \otimes v \otimes u$. The right side sends it through $v \otimes u \otimes w$ and $w \otimes u \otimes v$ to $w \otimes v \otimes u$, so the two sides agree. Example and non-example: let $R = A \otimes B$ for linear maps $A, B \colon V \to V$. Then $R_{12} R_{13} R_{23} = A^2 \otimes BA \otimes B^2$ and $R_{23} R_{13} R_{12} = A^2 \otimes AB \otimes B^2$. If $AB = BA$ the equation holds. If $A, B$ are invertible and $AB \ne BA$, it fails, because $A^2 \otimes (BA - AB) \otimes B^2 \ne 0$. Versions with a spectral parameter, $R(u)$ depending on $u \in \C$, are also called Yang–Baxter equations.