Every semisimple Lie algebra 
is classified by its Dynkin diagram. A Dynkin diagram is a
graph with a few different kinds of possible edges. The
connected components of the graph correspond
to the irreducible subalgebras of 
. So a simple Lie algebra's
Dynkin diagram has only one component. The rules are restrictive. In fact, there
are only certain possibilities for each component, corresponding to the classification
of semi-simple Lie algebras.
The roots of a complex Lie algebra form a lattice of rank 
in a Cartan subalgebra 
, where 
is the Lie algebra rank
of 
. Hence, the root
lattice can be considered a lattice in 
. 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 
. 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. An admissible diagram has no loops.
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 Coxeter-Dynkin diagram, also called a Coxeter graph, is the same as a Dynkin diagram, without the arrows, although sometimes these are also called Dynkin diagrams. The Coxeter diagram is sufficient to characterize the algebra, as can be seen by enumerating connected diagrams.
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.
