Hall algebras of weighted projective lines and quantum algebras

My talk is based on two joint papers with Olivier Schiffmann arXiv:0903.4828 and arXiv:1003.4412. The goal of our project is to apply the structure results on the derived category of coherent sheaves on a weighted projective line to a study of quantized enveloping algebras. In particular, we obtain a new realization of quantum affine algebras of simply-laced types, based on the orbifold quotients of P^1. We also get new results on certain quantum toroidal algebras, based on orbifolds of elliptic curves.