The classification of convex orders on affine root systems

We classify all total orders having a certain convex property on the positive root system of an arbitrary untwisted affine Lie algebra ${\frak g}$. Such total orders are called convex orders and are used to construct convex bases of Poincar\'e-Birkhoff-Witt type of the upper triangular subalgebra $U_q^+$ of the quantized enveloping algebra $U_q({\frak g})$.