  We study solutions to a generalized version of the classical Yang-Baxter
equation (CYBE) with values in a central simple Lie algebra over a field of
characteristic 0 from an algebro-geometric perspective. In particular, we
describe such solutions using cohomology free sheaves of Lie algebras on
projective curves. This framework leads to a new proof of the Belavin-Drinfeld
trichotomy, which asserts that over the ground field $\mathbb{C}$ any
non-degenerate solution of the CYBE is either elliptic, trigonometric or
rational. Furthermore, we obtain new structural results for solutions of the
generalized CYBE over arbitrary fields of characteristic 0. For example, we
prove that such solutions can be extended in an appropriate way to rational
maps on the product of two curves.
