Dynkin Diagram - Isomorphisms

Isomorphisms

Dynkin diagrams are conventionally numbered so that the list is non-redundant: for for for for and starting at The families can however be defined for lower n, yielding exceptional isomorphisms of diagrams, and corresponding exceptional isomorphisms of Lie algebras and associated Lie groups.

Trivially, one can start the families at or which are all then isomorphic as there is a unique empty diagram and a unique 1-node diagram. The other isomorphisms of connected Dynkin diagrams are:

These isomorphisms correspond to isomorphism of simple and semisimple Lie algebras, which also correspond to certain isomorphisms of Lie group forms of these. They also add context to the E_n family.