Extended Affine Weyl Groups

In this paper we study the Weyl groups of reduced extended affine root systems, the root systems of extended affine Lie algebras. We start by describing the extended affine Weyl group as a semidirect product of a finite Weyl group and a Heisenberg-like normal subgroup. This provides a unique expression for the Weyl group elements (in terms of some naturally arisen transformations) which is crucial in the further study of extended affine Weyl groups. We use this to give a presentation, called a presentation by conjugation, for an important subclass of extended affine Weyl groups. Using a new notion, called the index which is an invariant of the extended affine root systems, we show that one of the important features of finite and affine root systems (related to Weyl group) holds for the class of extended affine root systems.