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.