Root system

A (reduced) root system is a finite subset $\Phi\subset V\setminus\{0\}$ of a real finite-dimensional inner product space $(V,\langle\cdot,\cdot\rangle)$ such that:

1. $\Phi$ spans $V$.
2. If $\alpha\in\Phi$, then the only scalar multiples of $\alpha$ in $\Phi$ are $\pm\alpha$.
3. For each $\alpha\in\Phi$, the reflection
   $$s_\alpha(v)=v-\frac{2\langle v,\alpha\rangle}{\langle \alpha,\alpha\rangle}\,\alpha$$
   preserves $\Phi$, i.e. $s_\alpha(\Phi)=\Phi$.
4. (Integrality) For all $\alpha,\beta\in\Phi$,
   $$\frac{2\langle \beta,\alpha\rangle}{\langle \alpha,\alpha\rangle}\in\mathbb Z.$$

The subgroup of $\mathrm{GL}(V)$ generated by the reflections $\{s_\alpha\}$ is the Weyl group of $\Phi$.

Root systems arise from semisimple Lie algebras: if $\mathfrak g$ is semisimple and $\mathfrak h$ is a Cartan subalgebra, then the set of roots $\Phi\subset \mathfrak h^*$ (see root of a Lie algebra) becomes a root system after identifying $\mathfrak h^*$ with a real inner product space via the Killing form. The corresponding root space decomposition is the bridge between algebra and combinatorics.

Choosing a positive system picks out a basis of simple roots, from which one constructs the Cartan matrix and Dynkin diagram. This data underlies the classification of simple Lie algebras.