Extended Affine Weyl Groups

Abstract

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 Ž . 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 Ž . features of finite and affine root systems related to Weyl group holds for the class of extended affine root systems. ᮊ

On the Presentations of Extended Affine Weyl Groups

“…For completeness we have recorded parts (iii)-(iv) and (vii)-(viii) which can be found in [Az1] and [K]. The proofs of these parts are now easy consequences of our results in this section.…”

“…For the study of extended affine root systems and their Weyl groups we refer the reader to [Sa], [MS], [AABGP], [AzS2], [AzS1], [Az1], [Az2], [SaT], [Az4], [T2] and [T1].…”

Extended Affine Weyl Groups: Presentation by Conjugation via Integral Collection

“…• finite and affine Weyl groups, (see [St] and [MP,Proposition 5.3.3], • simply laced extended affine Weyl groups of rank > 1 (see [K, Theorem, III.1.14]), • extended affine Weyl groups of index zero, including extended affine Weyl groups of types F 4 and G 2 (see [A4,Theorem,5.15]), • nullity 2 extended affine Weyl groups of types A 1 , B ℓ , C ℓ (see [A3,Theorem 3.1]),…”

Módulos tipo Verma sobre álgebra TKK afim estendida

“…Essa seção foi baseada no trabalho [2] o qual introduziu a nomenclaturaálgebras de Lie afins estendidas. Introduziremos aqui os axiomas que definem umaálgebra de Lie afim estendida, estudaremos seu sistema de raízes, definiremos o tipo daálgebra e de seu sistema de raízes e o grupo de Weyl afim estendido [1].…”

“…De cada um deles obtemos distintos módulos tipo Verma, sendo um deles uma generalização do módulo de Verma clássico para o caso afim e o outro, a generalização do módulo de Verma imaginário. O principal resultado nesse capítuloé um critério de irredutibilidade: o módulo de Verma clássico sobre aálgebra TKK afim estendidaé irredutível se, e somente se, o módulo de Verma sobre sua subálgebra isomorfoàálgebra A (1) 1é irredutível. Mostramos também que á algebra TKK afim estendida não possui quocientes integráveis.…”