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

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 3, page 591-644
- ISSN: 1435-9855

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topBurban, Igor, and Henrich, Thilo. "Vector bundles on plane cubic curves and the classical Yang–Baxter equation." Journal of the European Mathematical Society 017.3 (2015): 591-644. <http://eudml.org/doc/277224>.

@article{Burban2015,

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 classication and prove that they are degenerations of the corresponding elliptic solutions.},

author = {Burban, Igor, Henrich, Thilo},

journal = {Journal of the European Mathematical Society},

keywords = {Yang–Baxter equations; elliptic fibrations; vector bundles on curves of genus one; derived categories; Massey products; Yang-Baxter equation; elliptic fibrations; vector bundles on curves of genus one; derived categories; Massey products},

language = {eng},

number = {3},

pages = {591-644},

publisher = {European Mathematical Society Publishing House},

title = {Vector bundles on plane cubic curves and the classical Yang–Baxter equation},

url = {http://eudml.org/doc/277224},

volume = {017},

year = {2015},

}

TY - JOUR

AU - Burban, Igor

AU - Henrich, Thilo

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

JO - Journal of the European Mathematical Society

PY - 2015

PB - European Mathematical Society Publishing House

VL - 017

IS - 3

SP - 591

EP - 644

AB - 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 classication and prove that they are degenerations of the corresponding elliptic solutions.

LA - eng

KW - Yang–Baxter equations; elliptic fibrations; vector bundles on curves of genus one; derived categories; Massey products; Yang-Baxter equation; elliptic fibrations; vector bundles on curves of genus one; derived categories; Massey products

UR - http://eudml.org/doc/277224

ER -

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