Where do root systems arise in mathematics?

Question

One often hears that root systems are ubiquitous in mathematics and physics. The most obvious occurrence of root systems is in the classification of complex simple Lie algebras. Where else do they arise? I would like to see a long list of areas in which they arise.

Answer 1

I first came across root systems in the classification of finite reflection groups. A point group $\Gamma\subseteq\mathrm O(\Bbb R^n)$ is a reflection group if it is generated by reflections at hyperplanes. It turns out that a set of reflections generates a finite group if and only if the normals of the reflection hyperplanes form a root system. Reflection groups in turn relate naturally to Coxeter groups, and so root systems also correspond to finite Coxeter groups.

Reflection groups are the most natural way to enumerating the known uniform polytopes (or more precisely, the Wythoffian polytops). Uniform polytopes fall into families indexed by root systems, and so this is a context in which root systems occur already from elementary geometric questions.

Another natural place to discover root systems via reflection groups is in crystallography

Answer 2

They arise in the representation theory of quivers: Gabriel's theorem says that a connected quiver has finite representation theory type if and only if it is of type ADE, and then the indecomposable representations correspond to the positive roots. This has been extended to symmetrizable Cartan types by Dlab and Ringel and to infinite root systems by Kac.

I think of this as ultimately deriving from the connection between root systems and crystalographic reflection groups (M Winter's answer): If $Q$ is a quiver, then the Grothedieck group of $Q$ representions is a lattice equipped with a bilinear form, and Bernstein, Gelfand and Ponomarev found "reflection functors" which act on this lattice by reflections.

Answer 3

The eigenvalue distribution functions of random matrices in different universality classes are determined by the multiplicities of the restricted roots of the corresponding symmetric spaces, see Random matrix theory and symmetric spaces by Caselle and Magnea.
For example, the Dyson index $\beta\in\{1,2,4\}$ equals the multiplicity $m_o$ of the ordinary roots of the restricted root lattice that characterizes the GOE, GUE, and GSE random-matrix ensembles.
This mathematics has applications in physics in the context of topological states of matter.