Yang-Baxter equation

Imagine three strands you want to braid past one another. You can move the first across the other two and then sort out the last pair, or sort the first pair and then move the third across — and physically you must reach the same braid either way. The Yang-Baxter equation is the exact algebraic statement that these two orders agree. It is the consistency condition that makes braidings, and the scattering of particles in integrable models, well-defined.

In its braided (constant) form, the equation is for an invertible map R : V ⊗ V -> V ⊗ V and reads R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12} as maps on V ⊗ V ⊗ V, where R_{ij} applies R to the i-th and j-th tensor factors and the identity elsewhere. Equivalently, writing the flip composed with R as a braiding c, the equation becomes the braid relation (c ⊗ id)(id ⊗ c)(c ⊗ id) = (id ⊗ c)(c ⊗ id)(id ⊗ c).

There is also a parameter-dependent version R_{12}(u) R_{13}(u+v) R_{23}(v) = R_{23}(v) R_{13}(u+v) R_{12}(u), central to exactly solvable lattice models and the algebraic Bethe ansatz, where u, v are spectral parameters. Solutions R are exactly the data that make module categories braided; quasitriangular Hopf algebras manufacture them in bulk via the universal R-matrix.