New solutions to the image-invariant Yang–Baxter equations at roots of unity: Cyclic representations Original Research Article

We find all the solutions to the image-invariant multi-parametric Yang–Baxter equations (YBE) at image defined on the cyclic (semi-cyclic, nilpotent) representations of the algebra. We derive the solutions in form of the linear combinations over the image-invariant objects — projectors. The direct construction of the projector operators at roots of unity gives us an opportunity to consider all the possible cases, including also degenerated one, when the number of the projectors becomes larger, and various types of solutions are arising, and as well as the inhomogeneous case. We give full classification of the YBE solutions for the considered representations. A specific character of the solutions is the existence of the arbitrary functions.