ELLIPTIC ROOT SYSTEMS OF TYPE A1, A COMBINATORIAL STUDY

We consider some combinatorics of elliptic root systems of type A1. In particular, with respect to a xed re ectable base, we give a precise description of the positive roots in terms of a positivity theorem. Also the set of reduced words of the corresponding Weyl group is precisely described. These then lead to a new characterization of the core of the corresponding Lie algebra, namely we show that the core is generated by positive root spaces.