Dynkin diagram

A Dynkin diagram records the angles and relative lengths of the simple roots. A multiple-edge arrow points toward the shorter root.

The connected Dynkin diagrams of finite reduced crystallographic root systems are $A_r$, $B_r$, $C_r$, $D_r$, $E_6$, $E_7$, $E_8$, $F_4$ and $G_2$. Bonds encode products of off-diagonal Cartan integers; multiple-bond arrows point toward shorter roots. The simply laced branching diagrams have arm lengths $(1,1,r-3)$ for $D_r$ and $(1,2,2)$, $(1,2,3)$, $(1,2,4)$ for the three exceptional $E$ types. The $B$ and $C$ chains have an end double bond, $F_4$ a middle double bond, and $G_2$ a triple bond. This edge weighting is distinct from the vertex labels used to classify nilpotent orbits.

The underlying graph of a finite Dynkin diagram is a forest. Normalize every simple root to length one. Each bonded pair has inner product at most $-1/2$, and every other distinct pair has nonpositive inner product. A cycle with $m$ vertices would make the squared norm of their sum at most $m-m=0$, contradicting their linear independence in an inner product space. Multiple bonds denote root-length information, not graph-theoretic parallel-edge cycles.

The finite Dynkin diagram $G_2$ has two vertices joined by a triple bond, with the arrow toward the short root. The squared root-length ratio is three. It represents the G2 root system.

The finite Dynkin diagram $F_4$ is a four-vertex chain whose central bond is double. The two consecutive long roots lie on one side, the two consecutive short roots on the other, and the arrow points toward the short pair. The remaining two bonds are single.

The finite Dynkin diagrams $E_6,E_7,E_8$ are simply laced trees with one trivalent vertex. Their three arm lengths, counting edges, are respectively $(1,2,2)$, $(1,2,3)$ and $(1,2,4)$.

For $n\ge 4$, the finite Dynkin diagram $D_n$ is a simply laced tree with one trivalent vertex and three arms containing $1,1,n-3$ edges. All its roots have the same length.

For $n\ge 2$, the finite Dynkin diagram $C_n$ is a chain with a terminal double bond. Its arrow points from the terminal long root toward the adjacent short root. It represents the Cn root system; $C_2$ is isomorphic to $B_2$ after relabelling.

For $n\ge 1$, the finite Dynkin diagram $A_n$ is a chain of $n$ vertices joined by single bonds. It represents the root system of ${\mathfrak{sl}}_{n+1}$.

An irreducible root system is simply laced when all roots have the same length, equivalently when its Dynkin diagram has only single edges. This is also equivalent towhenever $\alpha \ne \pm \beta$.

The connected positive-definite simply-laced Dynkin diagrams are $A_n$ for $n\ge 1$, $D_n$ for $n\ge 4$, and the exceptional diagrams $E_6,E_7,E_8$.

The $D_3$ roots are $\pm {\epsilon }_i\pm {\epsilon }_j$ for $1\le i<j\le 3$. The simple rootsform the three-node chain ${\alpha }_2-{\alpha }_1-{\alpha }_3$, so $D_3\cong A_3$.

The root-system isomorphism $D_3\cong A_3$ and the classification of complex simple Lie algebras give ${\mathfrak{so}}_6(\mathbb{C})\cong {\mathfrak{sl}}_4(\mathbb{C})$.

An extended, or affine, Dynkin diagram adjoins the root ${\alpha }_0=-\theta$, where $\theta$ is the highest root.