A Dynkin diagram is a graph with several possible kinds of edges that classifies a finite-dimensional
complex semisimple Lie algebra 
. The connected components
of the graph correspond to the simple
Lie algebras in the direct sum decomposition of
. A finite-dimensional
complex simple Lie algebra's Dynkin diagram
has only one connected component. The rules
are restrictive. In fact, there are only certain possibilities for each component,
corresponding to the classification of semisimple
Lie algebras.
The roots of a complex semisimple Lie algebra form a finite root
system in the dual vector space of a Cartan
subalgebra 
. Their integer
span is the root lattice, a lattice
of rank 
, where 
is the Lie algebra rank of 
. Its real span
can be identified with 
. A vertex,
or node, in the Dynkin diagram is drawn for each
Lie algebra simple root, which corresponds
to a generator of the root
lattice. Between two nodes 
and 
, an edge
is drawn if the simple roots are not perpendicular. One line
is drawn if the angle between them is 
, two lines if the
angle is 
, and three lines
are drawn if the angle is 
. The remaining possibility is the right
angle 
, for which no edge
is drawn. There are no other possible angles between Lie algebra simple roots. Alternatively, the
number of lines 
between the simple roots 
and 
is given by
 |
where 
is an entry in the Cartan matrix. In a Dynkin diagram,
an arrow is drawn from the longer root to the
shorter root (when the angle
is 
or 
).
The picture above shows the two simple roots for 
, at an angle of 
, in the root lattice.
Therefore, the Dynkin diagram for 
has two nodes, with three
lines between them.
Here are some properties of admissible Dynkin diagrams.
1. A diagram obtained by removing a node from an admissible diagram is admissible.
2. The underlying simple graph of an admissible diagram has no cycles.
3. No node has more than three lines attached to it.
4. A sequence of nodes with only two single lines can be collapsed to give an admissible diagram.
5. The only connected diagram with a triple line has two nodes.
A Dynkin diagram determines the Coxeter-Dynkin diagram of its Weyl group by forgetting the arrows
and encoding single, double, and triple edges by Coxeter
labels 3, 4, and 6. The arrows are needed to distinguish the Lie
algebras: for example, 
and 
have the same unoriented diagram but are nonisomorphic
for 
.
The simplest way to recover a simple Lie algebra from its Dynkin diagram is to first reconstruct its Cartan
matrix 
. The 
th node and 
th node are connected by 
lines.
Since 
iff 
, and otherwise 
, it is easy to find 
and 
, up to order, from their
product. The arrow in the diagram indicates which is
larger. For example, if node 1 and node
2 have two lines between them, from node
1 to node 2, then 
and 
.
However, it is worth pointing out that each simple Lie algebra can be constructed concretely. For instance, the infinite families
, 
, 
, and 
correspond to 
the special
linear Lie algebra, 
the odd orthogonal Lie algebra, 
the symplectic
Lie algebra, and 
the even orthogonal Lie algebra. The other simple
Lie algebras are called exceptional Lie
algebras, and have constructions related to the octonions.
