Root Systems and Dynkin Diagrams - Examples and Constructions

Explicit construction of root systems for classical Lie algebras provides concrete realization of the abstract theory. These examples demonstrate how Dynkin diagrams encode algebraic structure.

Example

Type $A_n$ ( $\mathfrak{sl}_{n+1}$ ):

The root system consists of vectors $e_i - e_j$ for $1 \leq i \neq j \leq n+1$ in $\mathbb{R}^{n+1}$ subject to $\sum e_i = 0$ . There are $n(n+1)$ roots total.

Simple roots: $\alpha_i = e_i - e_{i+1}$ for $i = 1, \ldots, n$

Dynkin diagram: $\circ - \circ - \cdots - \circ$ ( $n$ nodes)

The Weyl group is $W = S_{n+1}$ (symmetric group), acting by permuting coordinates.

Example

Type $B_n$ ( $\mathfrak{so}_{2n+1}$ ):

Root system: $\{\pm e_i \pm e_j : i \neq j\} \cup \{\pm e_i\}$ in $\mathbb{R}^n$

Simple roots: $\alpha_i = e_i - e_{i+1}$ ( $i < n$ ), $\alpha_n = e_n$

Dynkin diagram: $\circ - \circ - \cdots - \circ \Rightarrow \circ$ (arrow points to last node)

This system has short roots $\{\pm e_i\}$ and long roots $\{\pm e_i \pm e_j\}$ with length ratio $\sqrt{2}$ .

Example

Type $C_n$ ( $\mathfrak{sp}_{2n}$ ):

Root system: $\{\pm e_i \pm e_j : i \neq j\} \cup \{\pm 2e_i\}$ in $\mathbb{R}^n$

Simple roots: $\alpha_i = e_i - e_{i+1}$ ( $i < n$ ), $\alpha_n = 2e_n$

Dynkin diagram: $\circ - \circ - \cdots - \circ \Leftarrow \circ$ (arrow points from last node)

Here short roots are $\{\pm e_i \pm e_j\}$ and long roots are $\{\pm 2e_i\}$ .

Example

Type $D_n$ ( $\mathfrak{so}_{2n}$ ):

Root system: $\{\pm e_i \pm e_j : i \neq j\}$ in $\mathbb{R}^n$

Simple roots: $\alpha_i = e_i - e_{i+1}$ ( $i < n$ ), $\alpha_n = e_{n-1} + e_n$

Dynkin diagram: $\circ - \circ - \cdots - \begin{matrix}\circ \\ \circ\end{matrix}$ (fork at end)

All roots have the same length. The Weyl group includes sign changes and even permutations.

Remark

The exceptional root systems $G_2, F_4, E_6, E_7, E_8$ require more sophisticated constructions. For instance, $G_2$ can be realized in the plane as the root system of a hexagonal lattice with two lengths, while $E_8$ naturally lives in 8-dimensional space and has 240 roots.

Construction from Dynkin diagram: Given a Dynkin diagram, one can reconstruct the Cartan matrix $A = (a_{ij})$ where:

$a_{ii} = 2$
$a_{ij} a_{ji} =$ (number of edges between nodes $i$ and $j$ )
$|a_{ij}| \leq |a_{ji}|$ with equality iff roots have equal length

The Cartan matrix determines the root system up to isomorphism via the realization: $\alpha_j = \sum_i a_{ij} \lambda_i$ where $\{\lambda_i\}$ are fundamental weights (dual to simple coroots).

Example

Type $G_2$ Cartan matrix: $A = \begin{pmatrix} 2 & -1 \\ -3 & 2 \end{pmatrix}$ The off-diagonal entries multiply to 3, indicating a triple bond in the Dynkin diagram. The asymmetry $|-1| < |-3|$ indicates the arrow direction.

These constructions show that root systems, while abstractly defined, have concrete realizations as geometric configurations of vectors, making them amenable to both theoretical analysis and explicit computation.