My talk is based on two joint papers with Olivier Schiffmann arXiv:0903.4828 [4] and arXiv:1003.4412 [5]. 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.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/5312
[4] http://arxiv.org/abs/0903.4828
[5] http://arxiv.org/abs/1003.4412