A Criterion for Decomposabilty in QYBE

Abstract

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.

On derived-indecomposable solutions of the Yang--Baxter equation

“…Many authors focused on studying and classifying such solutions with a special emphasis on those which are also involutive and non-degenerate, i.e. solutions (X, r) such that r 2 = id, and if we write r(x, y) = (σ x (y), τ y (x)) then the maps σ x and τ y are bijective, for all x, y ∈ X; see [6,7,8,9,18,21,25,33,34,36,37,39]. On the other hand, almost nothing is known about indecomposable solutions which are non-degenerate and non-involutive (see [13,26,32]).…”