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In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these. The Weyl group of a semi-simple Lie group, a semi-simple Lie algebra, a semi-simple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra. It is named after Hermann Weyl. ==Weyl chambers== Removing the hyperplanes defined by the roots of Φ cuts up Euclidean space into a finite number of open regions, called Weyl chambers. These are permuted by the action of the Weyl group, and it is a theorem that this action is simply transitive. In particular, the number of Weyl chambers equals the order of the Weyl group. Any non-zero vector ''v'' divides the Euclidean space into two half-spaces bounding the hyperplane ''v''∧ orthogonal to ''v'', namely ''v''+ and ''v''−. If ''v'' belongs to some Weyl chamber, no root lies in ''v''∧, so every root lies in ''v''+ or ''v''−, and if α lies in one then −α lies in the other. Thus Φ+ := Φ∩''v''+ consists of exactly half of the roots of Φ. Of course, Φ+ depends on ''v'', but it does not change if ''v'' stays in the same Weyl chamber. The base of the root system with respect to the choice Φ+ is the set of ''simple roots'' in Φ+, i.e., roots which cannot be written as a sum of two roots in Φ+. Thus, the Weyl chambers, the set Φ+, and the base determine one another, and the Weyl group acts simply transitively in each case. The following illustration shows the six Weyl chambers of the root system A2, a choice of ''v'', the hyperplane ''v''∧ (indicated by a dotted line), and positive roots α, β, and γ. The base in this case is . image:Weyl chambers.png 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Weyl group」の詳細全文を読む スポンサード リンク
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