THE CLASSIFICATION OF CONVEX ORDERS ON AFFINE ROOT SYSTEMS

Abstract

We classify all total orders with a convex property on the positive root system of an arbitrary untwisted affine Lie algebra 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 universal enveloping algebra U_q (g).

Keywords:

• Convex order
• Affine root system
• Affine Weyl group
• Kac-Moody Lie algebra
• Quantum algebra
• Convex basis

ACKNOWLEDGMENTS

I thank Professor Akihiro Tsuchiya and Professor Takahiro Hayashi for constant help and precious advice.