Solving and classifying the solutions of the Yang-Baxter equation through a differential approach. Two-state systems

Understand

The formal derivatives of the Yang-Baxter equation with respect to its spectral parameters, evaluated at some fixed point of these parameters, provide us with two systems of differential equations.

• The derivatives of the $R$ matrix elements, however, can be regarded as independent variables and eliminated from the systems, after which two systems of polynomial equations are obtained in place.
• In general, these polynomial systems have a non-zero Hilbert dimension, which means that not all elements of the $R$ matrix can be fixed through them.
• Nonetheless, the remaining unknowns can be found by solving a few number of simple differential equations that arise as consistency conditions of the method.