Editing Dynkin diagram (section)

==Isomorphisms==
[[File:Dynkin Diagram Isomorphisms.svg|class=skin-invert-image|thumb|upright|The [[exceptional isomorphism]]s of connected Dynkin diagrams.]]
Dynkin diagrams are conventionally numbered so that the list is non-redundant: <math>n \geq 1</math> for <math>A_n,</math> <math>n \geq 2</math> for <math>B_n,</math> <math>n \geq 3</math> for <math>C_n,</math> <math>n \geq 4</math> for <math>D_n,</math> and <math>E_n</math> starting at <math>n=6.</math> The families can however be defined for lower ''n,'' yielding [[exceptional isomorphism]]s of diagrams, and corresponding exceptional isomorphisms of Lie algebras and associated Lie groups.

Trivially, one can start the families at <math>n=0</math> or <math>n=1,</math> 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:
* <math>A_1 \cong B_1 \cong C_1</math>
* <math>B_2 \cong C_2</math>
* <math>D_2 \cong A_1 \times A_1</math>
* <math>D_3 \cong A_3</math>
* <math>E_3 \cong A_1 \times A_2</math>
* <math>E_4 \cong A_4</math>
* <math>E_5 \cong D_5</math>

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 [[En (Lie algebra)|E<sub>n</sub> family]].<ref>{{citation |title=This Week's Finds in Mathematical Physics (Week 119) |date=April 13, 1998 |first=John |last=Baez |url=http://math.ucr.edu/home/baez/week119.html}}</ref>