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

Summary

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.

2000 Mathematics Subject Classifications: 16W30, 14F05, 18E30