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

### Igor Burban

Universität zu Köln, Germany### Thilo Henrich

Universität Bonn, Germany

## Abstract

In this article, we develop a geometric method to construct solutions of the classical Yang–Baxter equation, attaching a family of classical $r$-matrices to the Weierstrass family of plane cubic curves and a pair of coprime positive integers. 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.

## Cite this article

Igor Burban, Thilo Henrich, Vector bundles on plane cubic curves and the classical Yang–Baxter equation. J. Eur. Math. Soc. 17 (2015), no. 3, pp. 591–644

DOI 10.4171/JEMS/512