Abstract:
Let $G$ be a finite subgroup in $SU(2)$, and $Q$ the corresponding affine Dynkin diagram. We review the relation between the categories of $G$-equivariant sheaves on $P^1$ and $Rep Q_h$, where $h$ is an orientation of $Q$, constructing an explicit equivalence of corresponding derived categories. This allows one to construct the affine root system from the category of the equivariant sheaves. We also review the relation of this approach to recent work of Ocneanu on construction of simple Lie algebras in terms of the algebra of essential paths on $Q\times Z/hZ$.
