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.

Double soft theorem for generalised biadjoint scalar amplitudes

We study double soft theorem for the generalised biadjoint scalar field theory whose amplitudes are computed in terms of punctures on \mathbb{CP}^{k-1}ℂℙk−1. We find that whenever the double soft limit does not decouple into a product of single soft factors, the leading contributions to the double soft theorems come from the degenerate solutions, otherwise the non-degenerate solutions dominate. Our analysis uses the regular solutions to the scattering equations. Most of the results are presented for k=3k=3 but we show how they generalise to arbitrary kk. We have explicit analytic results, for any kk, in the case when soft external states are adjacent.