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

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 wa...

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Hauptverfasser:	Burban, Igor, Henrich, Thilo
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Schlagworte:	Mathematics - Algebraic Geometry Mathematics - Representation Theory
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creator	Burban, Igor Henrich, Thilo
description	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.
doi_str_mv	10.48550/arxiv.1202.5738
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title	Vector bundles on plane cubic curves and the classical Yang-Baxter equation
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