Oberseminar Algebra und Geometrie

Igor Burban  (Universität Paderborn)

Abstract:

Classical Yang-Baxter equation (CYBE) plays an important role in the modern theory of integrable systems. In a seminal work of Belavin and Drinfeld from the year 1983 it was proven that solutions of CYBE can have one of the following three types: elliptic, trigonometric or rational. Moreover, Belavin and Drinfeld gave a complete classification of all elliptic and trigonometric solutions.

In my talk, I am going to explain a geometric description of solutions of (CYBE). Namely, starting with any geometric datum (E, A), where E is a Weierstraß cubic curve and A a torsion free sheaf of Lie algebras (whose generic fiber is a given complex simple Lie algebra) with vanishing cohomology, one can associate to it in a canonical way a solution of CYBE. It turns out that all solutions of CYBE arise in this way. Moreover, the type of the curve in the datum (E, A) (i.e. smooth/nodal/cuspidal) determines the type of the corresponding solution of CYBE (i.e elliptic/trigonometric/rational).
The developed method will be illustrated by explicit examples. This talk is based on my joint works with Raschid Abedin, Lennart Galinat and Thilo Henrich.