Yang–Baxter equation

Summary

The Yang–Baxter equation is the identity $R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}$ for a linear map $R$ of $V \otimes V$. The indices say on which two of three tensor factors $R$ acts. It first appeared in exactly solvable models of statistical mechanics and in quantum scattering in one dimension. Composing with the flip turns a solution into a solution of the braid relation. So an invertible solution gives representations of the braid group. The universal R-matrix of a quasitriangular Hopf algebra satisfies it, and quantum groups are a rich source of solutions.