# Root Systems and Root Data

A (real) reflection is an automorphism of a real vector space that acts as negation on a one-dimensional subspace while fixing a hyperplane pointwise. The subspace is described by a vector called the *root*, while the hyperplane is described as the kernel of an element of the dual space called the *coroot*.

A root system is a collection of root/coroot pairs that is closed under the action of the corresponding reflections. Only finite root systems are supported at the present time. A root system gives a much more detailed description of a reflection representation of a finite Coxeter group.

Root systems are used to classify the *semisimple Lie algebras*. The closely related concept of a root datum is used to classify the *groups of Lie type*.

This is described in Chapters [Root Systems](../RootSystems/index-root-systems.md#chaprootsys) and [Root Data](../RootData/index-root-data.md#chaprootdtm).