# Extended Affine Root Systems III (Elliptic Weyl Groups)

### Kyoji Saito

Kyoto University, Japan### Tadayoshi Takebayashi

Waseda University, Tokyo, Japan

## Abstract

We give a presentation of an elliptic Weyl group *W*(*R*) (=the Weyl group for an elliptic root system*) *R*) in terms of the elliptic Dynkin diagram *Γ*(*R, G*) for the elliptic root system. The presentation is a generalization of a Coxeter system: the generators are in one to one correspondence with the vertices of the diagram and the relations consist of two groups : i) elliptic Coxeter relations attached to the diagram, and ii) a flniteness condition on the Coxeter transformation attached to the diagram. The group defined only by the elliptic Coxeter relations is isomorphic to the central extension *W*(*R*, *G*) of *W*(*R*) by an infinite cyclic group, called the hyperbolic extension of *W*(*R*). *) an elliptic root system=a 2-extended affine root system (see the introduction and the remark at its end).