Dynkin Diagram - Other Maps of Diagrams

Other Maps of Diagrams

Some additional maps of diagrams have meaningful interpretations, as detailed below. However, not all maps of root systems arise as maps of diagrams.

For example, there are two inclusions of root systems of A₂ in G₂, either as the six long roots or the six short roots. However, the nodes in the G₂ diagram correspond to one long root and one short root, while the nodes in the A₂ diagram correspond to roots of equal length, and thus this map of root systems cannot be expressed as a map of the diagrams.

Some inclusions of root systems can be expressed as one diagram being an induced subgraph of another, meaning "a subset of the nodes, with all edges between them". This is because eliminating a node from a Dynkin diagram corresponds to removing a simple root from a root system, which yields a root system of rank one lower. By contrast, removing an edge (or changing the multiplicity of an edge) while leaving the nodes unchanged corresponds to changing the angles between roots, which cannot be done without changing the entire root system. Thus, one can meaningfully remove nodes, but not edges. Removing a node from a connected diagram may yield a connected diagram (simple Lie algebra), if the node is a leaf, or a disconnected diagram (semisimple but not simple Lie algebra), with either two or three components (the latter for D_n and E_n). At the level of Lie algebras, these inclusions correspond to sub-Lie algebras.

The maximal subgraphs are ("conjugate" means "by a diagram automorphism"):

• A_n+1: A_n, in 2 conjugate ways.
• B_n+1: A_n, B_n.
• C_n+1: A_n, C_n.
• D_n+1: A_n (2 conjugate ways), D_n.
• E_n+1: A_n, D_n, E_n.
  • For E₆, two of these coincide: and are conjugate.
• F₄: B₃, C₃.
• G₂: A₁, in 2 non-conjugate ways (as a long root or a short root).

Finally, duality of diagrams corresponds to reversing the direction of arrows, if any: B_n and C_n are dual, while F₄, and G₂ are self-dual, as are the simply-laced ADE diagrams.