Root Systems and Dynkin Diagrams - Core Definitions

Root systems provide the combinatorial framework for classifying semisimple Lie algebras. They encode the structure of the Lie algebra in a finite configuration of vectors satisfying special symmetry properties.

Definition

Let $V$ be a finite-dimensional real vector space with inner product. A root system in $V$ is a finite set $\Phi \subset V$ of nonzero vectors (called roots) satisfying:

$\Phi$ spans $V$
If $\alpha \in \Phi$ , the only scalar multiples of $\alpha$ in $\Phi$ are $\pm \alpha$
For each $\alpha \in \Phi$ , the reflection $s_\alpha(v) = v - \frac{2\langle v, \alpha\rangle}{\langle \alpha, \alpha\rangle}\alpha$ maps $\Phi$ to itself
For $\alpha, \beta \in \Phi$ , the number $\frac{2\langle \beta, \alpha\rangle}{\langle \alpha, \alpha\rangle}$ is an integer

The fourth condition is called the crystallographic condition. It ensures that root systems arise from Lie algebras rather than arbitrary reflection groups.

Example

For $\mathfrak{sl}_3(\mathbb{C})$ with Cartan subalgebra $\mathfrak{h}$ of diagonal matrices, the root system consists of six roots in $\mathbb{R}^2$ : $\Phi = \{\pm \alpha_1, \pm \alpha_2, \pm(\alpha_1 + \alpha_2)\}$ where $\alpha_1, \alpha_2$ are simple roots. This forms a hexagonal configuration (type $A_2$ ).

Definition

The Weyl group $W$ of a root system $\Phi$ is the group generated by all root reflections $s_\alpha$ for $\alpha \in \Phi$ . This is a finite reflection group acting on $V$ .

The Weyl group is central to both the geometry of root systems and the representation theory of the associated Lie algebra.

Definition

A subset $\Delta = \{\alpha_1, \ldots, \alpha_n\} \subseteq \Phi$ is a set of simple roots if:

$\Delta$ is a basis of $V$
Every root $\beta \in \Phi$ can be written as $\beta = \sum_i k_i \alpha_i$ where all $k_i \geq 0$ (positive root) or all $k_i \leq 0$ (negative root)

Simple roots provide a canonical basis for describing the root system. Every root system has a simple root system, unique up to Weyl group action.

Remark

The Cartan matrix $A = (a_{ij})$ is defined by $a_{ij} = \frac{2\langle \alpha_j, \alpha_i\rangle}{\langle \alpha_i, \alpha_i\rangle}$ for simple roots $\{\alpha_i\}$ . This matrix completely determines the root system and hence the Lie algebra structure.

Definition

The Dynkin diagram of a root system with simple roots $\Delta = \{\alpha_1, \ldots, \alpha_n\}$ is a graph with:

One node for each simple root
Nodes $i$ and $j$ connected by $a_{ij} \cdot a_{ji}$ edges (where $a_{ij}$ are Cartan matrix entries)
An arrow pointing from longer root to shorter root if roots have different lengths

Dynkin diagrams provide a visual classification of root systems and semisimple Lie algebras. The simply-laced diagrams (no multiple edges) correspond to types $A_n, D_n, E_6, E_7, E_8$ .

Example

Type $A_n$ has Dynkin diagram: $\circ - \circ - \cdots - \circ$ ( $n$ nodes in a line) Type $D_n$ has a fork at one end: $\circ - \circ - \cdots - \begin{matrix}\circ \\ \circ\end{matrix}$ Type $G_2$ has a triple edge: $\circ \equiv \circ$ with arrow

The classification theorem states that connected Dynkin diagrams are exactly the types $A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2$ , corresponding to all simple Lie algebras over $\mathbb{C}$ .