A Criterion for Decomposabilty in QYBE

In this paper, set theoretic solutions of the Quantum Yang-Baxter Equations are considered. Etingof et al. [8] defined the structure group for non-degenerate solutions and gave some properties of this group. In particular, they provided a criterion for decomposability of involutive solutions based on the transitivity of the structure group. In that paper, the diagonal permutation T is also introduced. It is known that this permutation is trivial exactly when the solution is square free. Rump [12] proved that these solutions are decomposable except in the trivial case. Later, Ramirez and Vendramin [11] gave some criteria for decomposability related with the diagonal permutation T. In this paper it was proven that an involutive solution is decomposable when the number of symbols of the solution and the order of the diagonal permutation T are coprime.