Dynkin diagram

They have the following correspondence for the Lie algebras associated to classical groups over the complex numbers:

For the exceptional groups, the names for the Lie algebra and the associated Dynkin diagram coincide.

Lastly, sometimes associated objects are referred to by the same notation, though this cannot always be done regularly. Examples include:

These latter notations are mostly used for objects associated with exceptional diagrams – objects associated to the regular diagrams (A, B, C, D) instead have traditional names.

Dynkin diagrams must satisfy certain constraints; these are essentially those satisfied by finite Coxeter–Dynkin diagrams, together with an additional crystallographic constraint.

Dynkin diagrams differ from Coxeter diagrams of finite groups in two important respects:

At the level of root systems the direction corresponds to pointing towards the shorter vector; edges labeled "3" have no direction because the corresponding vectors must have equal length. (Caution: Some authors reverse this convention, with the arrow pointing towards the longer vector.)
Dynkin diagrams must satisfy an additional restriction, namely that the only allowable edge labels are 2, 3, 4, and 6, a restriction not shared by Coxeter diagrams, so not every Coxeter diagram of a finite group comes from a Dynkin diagram.
At the level of root systems this corresponds to the crystallographic restriction theorem, as the roots form a lattice.

The corresponding mathematical objects classified by the diagrams are:

The blank in the upper right, corresponding to directed graphs with underlying undirected graph any Coxeter diagram (of a finite group), can be defined formally, but is little-discussed, and does not appear to admit a simple interpretation in terms of mathematical objects of interest.

There are natural maps down – from Dynkin diagrams to undirected Dynkin diagrams; respectively, from root systems to the associated Weyl groups – and right – from undirected Dynkin diagrams to Coxeter diagrams; respectively from Weyl groups to finite Coxeter groups.

Disconnected diagrams, which correspond to semisimple Lie algebras, may have automorphisms from exchanging components of the diagram.

Diagram automorphisms in turn yield additional Lie groups and groups of Lie type, which are of central importance in the classification of finite simple groups.

Folding can be applied to reduce questions about (semisimple) Lie algebras to questions about simply-laced ones, together with an automorphism, which may be simpler than treating multiply laced algebras directly; this can be done in constructing the semisimple Lie algebras, for instance. See for further discussion.

The maximal subgraphs are as follows; subgraphs related by a diagram automorphism are labeled "conjugate":

Alternative convention include writing a number by the edge to indicate multiplicity (commonly used in Coxeter diagrams), darkening nodes to indicate root length, or using 120° angles on valence 2 nodes to make the nodes more distinct.