# Publications of the Research Institute for Mathematical Sciences

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**Volume 33, Issue 2, 1997, pp. 301–329**

**DOI: 10.2977/prims/1195145453**

Published online: 1997-04-30

Extended Affine Root Systems III (Elliptic Weyl Groups)

Kyoji Saito^{[1]}and Tadayoshi Takebayashi

^{[2]}(1) Kyoto University, Japan

(2) Waseda University, Tokyo, Japan

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).

*Keywords: *

Saito Kyoji, Takebayashi Tadayoshi: Extended Affine Root Systems III (Elliptic Weyl Groups). *Publ. Res. Inst. Math. Sci.* 33 (1997), 301-329. doi: 10.2977/prims/1195145453