Lie Groups pp 129-144

# The Root System

• Daniel Bump
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 225)

## Abstract

A Euclidean space is a real vector space $$\mathcal{V}$$ endowed with an inner product, that is, a positive definite symmetric bilinear form. We denote this inner product by $$\left \langle \;,\;\right \rangle$$. If $$0\neq \alpha \in \mathcal{V}$$, consider the transformation $$s_{\alpha }: \mathcal{V}\longrightarrow \mathcal{V}$$ given by
$$\displaystyle{ s_{\alpha }(x) = x -\frac{2\left \langle x,\alpha \right \rangle } {\left \langle \alpha,\alpha \right \rangle } \alpha. }$$
(18.1)
This is the reflection attached to α. Geometrically, it is the reflection in the plane perpendicular to α. We have $$s_{\alpha }(\alpha ) = -\alpha$$, while any element of that plane (with $$\left \langle x,\alpha \right \rangle = 0$$) is unchanged by s α.

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