Igor Burban (Mainz)
Title: Vector bundles on cubic curves and Yang-Baxter equations
Abstract: in my talk based on a joint work with Bernd Kreussler I
am going to speak about applications of the theory of
vector bundles on elliptic curves and their degenerations
to the classical and quantum Yang-Baxter equations.
This unexpected connection was recently discovered by
Polishchuk in a framework of the homological mirror
symmetry and his study of A-infinity structures on derived
categories of coherent sheaves.
By a classical result of Belavin and Drinfeld there are
three types of solutions of the classical Yang-Baxter
equations: elliptic, trigonometric and rational.
It turns out that this trichotomy corresponds exactly
to three types of projective curves with trivial
canonical bundle: elliptic curves, Kodaira cycles and
curves with more complicated singularities like a cuspidal curve.
We carry out some explicit calculations of r-matrices and
their quantizations coming from singular curves. This approach
gives a new insight to a study of degenerations of solutions of the
classical and quantum Yang-Baxter equations.