Simply Laced Extended Affine Weyl Groups (A Finite Presentation)

Abstract

Extended affine Weyl groups are the Weyl groups of extended affine root systems. Finite presentations for extended affine Weyl groups are known only for nullities ≤ 2, where for nullity 2 there is only one known such presentation. We give a finite presentation for the class of simply laced extended affine Weyl groups. Our presentation is nullity free if rank > 1 and for rank 1 it is given for nullities ≤ 3. The generators and relations are given uniformly for all types, and for a given nullity they can be read from the corresponding finite Cartan matrix and the semilattice involved in the structure of the root system. §0. Introduction Extended affine Weyl groups are the Weyl groups of extended affine root systems which are higher nullity generalizations of affine and finite root systems. In 1985, K. Saito [S] introduced axiomatically the notion of an extended affine root system, he was interested in possible applications to the study of singularities.Extended affine root systems also arise as the root systems of a class of infinite dimensional Lie algebras called extended affine Lie algebras. A systematic study of extended affine Lie algebras and their root systems is given in

Presentation by conjugation for A1-type extended affine Weyl groups

“…For a more detailed exposition, we refer the reader to [AABGP] and [AS1], in particular we will use the notation and concepts introduced there without further explanations.…”

“…For the study of extended affine root systems and extended affine Weyl groups we refer the reader to [MS,Sa,AABGP,A1,A2,A3,A4,SaT,T,AS1,AS2]. The authors would like to thank A. Abdollahi for a fruitful discussion which led to the proof of crucial Lemma 5.1.…”

“…These notions naturally arise in the study of extended affine Weyl groups and their presentations. In Sections 2 and 3, we record some important results mainly from [AS1] which are crucial for the up coming sections, and establish a few preliminary results regarding extended affine Weyl groups.…”

Extended affine Weyl groups of type A 1

“…In this work we consider type A 1 and complete Takebayashi's result for simply laced nullity 3 extended affine Weyl groups. To achieve this, we have employed a new method using a recently given finite presentation for reduced extended affine Weyl groups (see [6] and [7]). This approach considers nullities ν ≤ 3 at once, in particular it provides a new proof for the elliptic Weyl group of type A 1 .…”

Presentation by conjugation for A1-type extended affine Weyl groups

“…(i) From (2.8),[AS1, Lemma 2.8] and the way z J is defined we havew α i +τ J w α i = T τ J α i = T r∈J σ…”