Torsion free sheaves on Weierstrass cubic curves and the classical Yang-Baxter equation

TL;DR

This paper develops an algebro-geometric framework using torsion free sheaves on Weierstrass cubic curves to study solutions of the classical Yang-Baxter equation, linking algebraic geometry with integrable systems.

Contribution

It introduces a novel approach connecting torsion free sheaves on cubic curves to solutions of the classical Yang-Baxter equation, expanding the geometric understanding of these solutions.

Findings

Established a correspondence between torsion free sheaves and Yang-Baxter solutions

Provided new geometric constructions for classical r-matrices

Extended the theory to Weierstrass cubic curves

Abstract

This work deals with an algebro-geometric theory of solutions of the classical Yang-Baxter equation based on torsion free coherent sheaves of Lie algebras on Weierstrass cubic curves.