Root Systems and Dynkin Diagrams - Key Properties

Root systems possess remarkable properties that make them perfect tools for classification. Understanding these properties reveals why root systems provide such a complete description of semisimple Lie algebras.

Theorem

Root System Properties

For a root system $\Phi$ in $V$ :

If $\alpha, \beta \in \Phi$ with $\alpha + \beta \neq 0$ , then $\frac{2\langle \beta, \alpha\rangle}{\langle \alpha, \alpha\rangle} \in \{0, \pm 1, \pm 2, \pm 3\}$
If $\alpha, \beta \in \Phi$ , then $\beta - \frac{2\langle \beta, \alpha\rangle}{\langle \alpha, \alpha\rangle}\alpha \in \Phi$
The Weyl group $W$ acts transitively on roots of the same length
All roots have at most two different lengths, with ratio $\sqrt{2}$ or $\sqrt{3}$

These constraints severely limit possible root system configurations, enabling complete classification.

Example

For type $B_2$ , there are 4 short roots and 4 long roots with length ratio $\sqrt{2}$ . The Weyl group $W \cong D_4$ (dihedral group of order 8) acts transitively on short roots and on long roots separately.

Definition

A root system is reducible if $\Phi = \Phi_1 \cup \Phi_2$ where $\Phi_1 \perp \Phi_2$ (orthogonal subsets). Otherwise it is irreducible. Every root system decomposes uniquely into irreducible components.

Remark

Irreducible root systems correspond to simple Lie algebras, while reducible systems correspond to direct sums. The classification problem reduces to classifying irreducible root systems.

The positive roots $\Phi^+ = \{\alpha \in \Phi : \alpha = \sum k_i \alpha_i, k_i > 0\}$ form exactly half of $\Phi$ , with $\Phi = \Phi^+ \cup (-\Phi^+)$ . Simple roots are precisely the indecomposable positive roots.

Theorem

Weyl Chamber Decomposition

The hyperplanes $H_\alpha = \{v \in V : \langle v, \alpha\rangle = 0\}$ for $\alpha \in \Phi$ divide $V$ into finitely many regions called Weyl chambers. The Weyl group acts simply transitively on Weyl chambers.

The fundamental Weyl chamber $C = \{v \in V : \langle v, \alpha_i\rangle > 0 \text{ for all simple } \alpha_i\}$ is particularly important. Every vector in $V$ is $W$ -equivalent to a unique vector in $\overline{C}$ .

Example

For $A_2$ , the six root hyperplanes divide the plane into 12 triangular Weyl chambers. The Weyl group $W \cong S_3$ permutes these chambers, corresponding to the 6 elements of $S_3$ acting on the root system.

Height function: For $\alpha = \sum k_i \alpha_i \in \Phi^+$ , define $\text{ht}(\alpha) = \sum k_i$ . The unique root of maximal height is called the highest root, denoted $\tilde{\alpha}$ . It plays a special role in representation theory and affine Lie algebras.

Remark

The Cartan integers $n_{\alpha,\beta} = \frac{2\langle \alpha, \beta\rangle}{\langle \alpha, \alpha\rangle}$ determine the action of the Lie algebra: $[E_\alpha, E_\beta] = N_{\alpha,\beta} E_{\alpha+\beta}$ when $\alpha + \beta \in \Phi$ , where structure constants $N_{\alpha,\beta}$ can be normalized using root string theory.

These properties show that root systems encode both the algebraic structure (via Cartan integers) and geometric structure (via Weyl chambers and reflections) of semisimple Lie algebras.