Weyl Groups Associated with Affine Reflection Systems of Type $A_1$ (Coxeter Type Defining Relations)

Abstract

We offer a presentation for the Weyl group of an affine reflection system R of type A1 as well as a presentation for the so called hyperbolic Weyl group associated with an affine reflection system of type A1. Applying these presentations to extended affine Weyl groups, and using a description of the center of the hyperbolic Weyl group, we also give a new finite presentation for an extended affine Weyl group of type A1. Our presentation for the (hyperbolic) Weyl group of an affine reflection system of type A1 is the first nontrivial presentation given in this generality, and can be considered as a model for other types.2010 Mathematics Subject Classification: Primary 20F55; Secondary 17B67. Keywords: affine reflection system, Weyl group, hyperbolic Weyl group, extended affine Weyl group. §0. Introduction Weyl groups, as reflection groups, always give a geometric meaning to underlying structures such as root systems, Lie algebras and Lie groups. Thus to get a "good" perspective of these structures, one needs to have a better understanding of their Weyl groups. The present work is dedicated to the study of some new presentations for (hyperbolic) Weyl groups associated with affine reflection systems of type A 1 .Affine reflection systems are the most general known extensions of finite and affine root systems introduced by E. Neher and O. Loos in [LN2]. They include Communicated by M. Kashiwara. Received July 28, 2011. Revised January 8, 2012, March 14, 2012, and April 16, 2012 In [AYY], the authors introduce an equivalent definition for an affine reflection system (see Definition 1.1) which we will use here. In the finite and affine cases, the corresponding Weyl groups are essentially known. In particular, it is known that they are Coxeter groups and that their actions implement a specific geometric and combinatorial structure on their underlying root systems. In the extended affine case, however, it is known that if the nullity is greater than one, then the corresponding hyperbolic Weyl groups, called extended affine Weyl groups, are not Coxeter groups (see [H3, Theorem 3.6] In the study of groups associated with affine reflection systems and other extensions of finite and affine root systems, type A 1 has always played a special role and is usually considered as a model for other types. A philosophical justification for this is that any (tame) affine reflection system can be considered as the union of a family of affine reflection systems of type A 1 .In this work, we study two groups associated with an affine reflection system R. Let R be an affine reflection system of type A 1 in an abelian group A. In Section 1, we give preliminary definitions as well as record some results and facts on affine reflection systems. In Sections 2 and 3, we offer two presentations for W and W, respectively (see Theorems 2.6 and 3.5). A quick look at these presentations shows that both W and W have soluble word problems. Also, using the description of the center of W given in Proposition 3.6, one perceives that W is a centr...