Dynkin diagram

A graph encoding the angles and relative lengths among simple roots of a semisimple Lie algebra via the Cartan matrix.

Dynkin diagram

Let $\mathfrak g$ be a complex semisimple Lie algebra . Fix a Cartan subalgebra $\mathfrak h$ and a choice of positive roots in its root system . Let $\Delta=\{\alpha_1,\dots,\alpha_\ell\}$ be the corresponding set of simple roots .

Definition

The Cartan matrix $A=(a_{ij})$ (see Cartan matrix ) is defined by

a_{ij} := \langle \alpha_i^\vee,\alpha_j\rangle,

where $\alpha_i^\vee$ is the coroot associated to $\alpha_i$ .

The Dynkin diagram of $(\mathfrak g,\Delta)$ is the graph with:

one vertex for each simple root $\alpha_i$ ;
between vertices $i$ $i$ and $j$ $j$ :
- no edge if $a_{ij}a_{ji}=0$ ,
- a single, double, or triple edge if $a_{ij}a_{ji}=1,2,3$ respectively;
if roots have different lengths, the multiple edge is oriented toward the shorter root (equivalently, toward the vertex with larger $|a_{ij}|$ ).

Up to isomorphism, the Dynkin diagram does not depend on choices beyond the isomorphism class of $\mathfrak g$ .

Why it matters

The diagram packages the essential combinatorics of the root system (angles and length ratios) into a finite graph. The connected Dynkin diagrams classify complex simple Lie algebras ; see classification of simple Lie algebras . This is the starting point for describing highest-weight representation theory.