Yang-Baxter R-matrix ($R\in \mathrm{GL}(V\otimes V)$)

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.