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

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

2012-02-26

arxiv.org/abs/1202.5738v2

Mathematics

Algebraic Geometry

50 pages, a new section on elliptic solutions is included

Scientific paper

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.

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