Yang-Baxter equation ($R_{12}(u)R_{13}(u+v)R_{23}(v)=R_{23}(v)R_{13}(u+v)R_{12}(u)$)

For a two-body operator $R(u)$ acting on a tensor product of particle species spaces, the spectral equation is $R_{12}(u)R_{13}(u+v)R_{23}(v)=R_{23}(v)R_{13}(u+v)R_{12}(u)$. It guarantees agreement between the two ways to reorder three particles in factorized scattering. A Yang–Baxter operator is a related braid-form operator; multiplying by the permutation operator converts between the braided and unbraided conventions.

An invertible linear map on $V\otimes V$ is an R-matrix when it obeys $R_{12}{R}_{13}{R}_{23}=R_{23}{R}_{13}{R}_{12}$ on $V^{\otimes 3}$. Here $R_{13}$ acts on the first and third factors. The flipped operator $PR$ obeys the braid relation. This numerical-matrix notion is distinguished from a universal quasitriangular element in a Hopf algebra tensor square.