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

Burban, 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 -