Dynkin Diagram - Related Classifications

Related Classifications

Dynkin diagrams can be interpreted as classifying many distinct, related objects, and the notation "A_n, B_n, ..." is used to refer to all such interpretations, depending on context; this ambiguity can be confusing.

The central classification is that a simple Lie algebra has a root system, to which is associated an (oriented) Dynkin diagram; all three of these may be referred to as B_n, for instance.

The unoriented Dynkin diagram is a form of Coxeter diagram, and corresponds to the Weyl group, which is the finite reflection group associated to the root system. Thus B_n may refer to the unoriented diagram (a special kind of Coxeter diagram), the Weyl group (a concrete reflection group), or the abstract Coxeter group.

Note that while the Weyl group is abstractly isomorphic to the Coxeter group, a specific isomorphism depends on an ordered choice of simple roots. Beware also that while Dynkin diagram notation is standardized, Coxeter diagram and group notation is varied and sometimes agrees with Dynkin diagram notation and sometimes does not.

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

• The root lattice generated by the root system, as in the E₈ lattice. This is naturally defined, but not one-to-one – for example, A₂ and G₂ both generate the hexagonal lattice.
• An associated polytope – for example Gosset 4₂₁ polytope may be referred to as "the E₈ polytope", as its vertices are derived from the E₈ root system and it has the E₈ Coxeter group as symmetry group.
• An associated quadratic form or manifold – for example, the E₈ manifold has intersection form given by the E₈ lattice.

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.

The index (the n) equals to the number of nodes in the diagram, the number of simple roots in a basis, the dimension of the root lattice and span of the root system, the number of generators of the Coxeter group, and the rank of the Lie algebra. However, n does not equal the dimension of the defining module (a fundamental representation) of the Lie algebra – the index on the Dynkin diagram should not be confused with the index on the Lie algebra. For example, corresponds to which naturally acts on 9-dimensional space, but has rank 4 as a Lie algebra.

The simply laced Dynkin diagrams, those with no multiple edges (A, D, E) classify many further mathematical objects; see discussion at ADE classification.