根系統和Weyl群的軌跡

Abstract

令g為一個有限維的複半單李代數，而h是g的Cartan子李代數。 則g會導出一個包含許多根的根系統。每個根沿著它自己的超平面又可導出一個根反射。 這些根反射所生成的群叫做Weyl群，這個群在h*有群作用。現在給定任意兩個h*的向量， 我們的目標是藉由觀察Weyl群的結構，找出一個有系統的方法去判斷這兩個向量是否在同一個Weyl群的軌跡裡。 對於An,Bn,Cn,Dn,G2型態的李代數，我們觀察Weyl群作用在歐氏空間的行為。 對於F4型態的李代數，觀察F4的根系統的自同構與D4的根系統的自同構之間的關係， 並藉此用D4的Weyl群去描述F4的Weyl群。

Let g be a finite dimensional complex semisimple Lie algebra with the Cartan subalgebra h. g induces a root system containing roots. Each root gives a reflection with respect to its hyperplane. These reflections generate a group W called Weyl group acting on on h*. Given two vectors, our purpose is to find a systematic method to judge if they are in the sane W-orbit by observing the structure of W. For type An,Bn,Cn,Dn,G2, we study the W-action on Euclidean space. For type F4, observe the relation between the automorphism of the root system of F4 and it of D4. Then describe the Weyl group of F4 by the Weyl group of D4.