New solutions to the slq(2)-invariant Yang-Baxter equations at roots of unity: Cyclic representations

We find all the solutions to the slq(2)-invariant multi-parametric Yang-Baxter equations (YBE) at q=i defined on the cyclic (semi-cyclic, nilpotent) representations of the algebra. We derive the solutions in form of the linear combinations over the slq(2)-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.