On the solutions to the multi-parametric Yang–Baxter equations

A unified approach is applied in the consideration of the multi-parametric (colored) Yang–Baxter equations (YBE) and the usual YBE with two-parametric R-matrices, relying on the existence of the arbitrary functions in the general solutions. The colored YBE are considered with the R  -matrices defined on two and three-dimensional states. We present an exhaustive study and the overall solutions for the YBE with 4×44×4 colored R  -matrices. The established classification includes new multi-parametric free fermionic solutions. In the context of the given approach there are obtained the colored solutions to the YBE with 9×99×9R-matrices having 15 non-zero elements.