Geometrization of trigonometric solutions of the associative and classical Yang-Baxter equations

TL;DR

This paper presents a geometric framework for constructing all nondegenerate trigonometric solutions to the associative and classical Yang-Baxter equations, linking solutions to spherical orders and sheaves over a genus 1 curve.

Contribution

It introduces a unified geometric approach to derive all such solutions from spherical structures on a nodal curve of genus one.

Findings

All solutions are obtained from spherical orders and sheaves over the same curve.

Provides a geometric classification of solutions to the Yang-Baxter equations.

Connects algebraic solutions to geometric objects on algebraic curves.

Abstract

We describe a geometric construction of all nondegenerate trigonometric solutions of the associative and classical Yang-Baxter equations. In the associative case the solutions come from symmetric spherical orders over the irreducible nodal curve of arithmetic genus $1$, while in the Lie case they come from spherical sheaves of Lie algebras over the same curve.