The classification of convex orders on affine root systems

Abstract

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

30pages, AMS-LaTeX