Vector bundles on degenerations of elliptic curves and Yang-Baxter equations [Elektronische Ressource] / Thilo Grunwald-Henrich. Mathematisch-Naturwissenschaftliche Fakultät

Vector bundles on degenerations of elliptic curves andYang-Baxter equationsDissertationzurErlangung des Doktorgrades (Dr. rer. nat.)derMathematisch-Naturwissenschaftlichen FakultätderRheinischen Friedrich-Wilhelms-Universität Bonnvorgelegt vonThilo Grunwald-HenrichausBonnBonn 2011Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultät derRheinischen Friedrich-Wilhelms-Universität Bonn1. Gutachter: PD Dr. Igor Burban2. Gutachter: Prof. Dr. Catharina StroppelTag der Promotion: 25.10.2011Erscheinungsjahr: 2011SummaryIn this thesis, we study connections between vector bundles on degenerations of ellipticcurvesandtheclassical,quantumandassociativeYang-Baxterequation. Letg =sl (C)nand let U denote the universal enveloping algebra of g. The classical Yang-Baxterequation (CYBE) is given as follows:! " ! " ! "12 23 12 13 13 23r (y ,y),r (y ,y) + r (y ,y),r (y ,y) + r (y ,y),r (y ,y) =0,1 2 2 3 1 2 1 3 1 3 2 32 ijwhere r :C ! g"g is a meromorphic function and r (y,y):g"g!U"U"U isi jthe embedding given by (i,j). This equation plays an important role in mathematicalphysics, representation theory and integrable systems.In 1982, Belavin and Drinfeld gave a classification of solutions of the CYBE. In par-ticular, they proved that any solution of the CYBE is either elliptic, trigonometric orrational. Moreover, they described all elliptic and trigonometric solutions.