Massey Products on Cycles of Projective Lines and Trigonometric Solutions of the Yang–Baxter Equations

Abstract

We show that a nondegenerate unitary solution r(u, v) of the associative Yang–Baxter equation (AYBE) for \(\mathrm{Mat}(N,\mathcal{C})\) (see [7]) with the Laurent series at u = 0 of the form r(u, v) = \(\frac{1 \otimes 1}{u}\) + r 0(v) + ⋯ satisfies the quantum Yang–Baxter equation, provided the projection of r 0(v) to sl N ⊗ sl N has a period. We classify all such solutions of the AYBE, extending the work of Schedler [8]. We also characterize solutions coming from triple Massey products in the derived category of coherent sheaves on cycles of projective lines.