Vector bundles on plane cubic curves and the classical Yang-Baxter equation

TL;DR

This paper introduces a geometric approach to constructing solutions of the classical Yang-Baxter equation using plane cubic curves, revealing new elliptic and rational r-matrices and their degenerations.

Contribution

It develops a geometric method linking plane cubic curves to classical r-matrices, unifying elliptic and rational solutions and describing their degenerations.

Findings

All elliptic r-matrices come from smooth cubic curves.

Solutions for cuspidal cubic are rational and explicitly computed.

Degenerations of elliptic solutions are described in terms of Stolin's classification.

Abstract

In this article, we develop a geometric method to construct solutions of the classical Yang-Baxter equation, attaching to the Weierstrass family of plane cubic curves and a pair of coprime positive integers, a family of classical r-matrices. It turns out that all elliptic r-matrices arise in this way from smooth cubic curves. For the cuspidal cubic curve, we prove that the obtained solutions are rational and compute them explicitly. We also describe them in terms of Stolin's classification and prove that they are degenerations of the corresponding elliptic solutions.