Dynkin Diagram - Affine Dynkin Diagrams

For more details on this topic, see Affine root system.

There are extensions of Dynkin diagrams, namely the affine Dynkin diagrams; these classify Cartan matrices of affine Lie algebras. These are classified in (Kac 1994, Chapter 4, pp. 47–), specifically listed on (Kac 1994, pp. 53–55). Affine diagrams are denoted as or where X is the letter of the corresponding finite diagram, and the exponent depends on which series of affine diagrams they are in. The first of these, are most common, and are called extended Dynkin diagrams and denoted with a tilde, and also sometimes marked with a + superscript. as in . The (2) and (3) series are called twisted affine diagrams.

See Dynkin diagram generator for diagrams.


The set of extended affine Dynkin diagrams, with added nodes in green
"Twisted" affine forms are named with (2) or (3) superscripts.
(k is the number of yellow nodes in the graph)

Here are all of the Dynkin graphs for affine groups up to 10 nodes. Extended Dynkin graphs are given as the ~ families, the same as the finite graphs above, with one node added. Other directed-graph variations are given with a superscript value (2) or (3), representing foldings of higher order groups. These are categorized as Twisted affine diagrams.

Connected affine Dynkin graphs up to (2 to 10 nodes)
(Grouped as undirected graphs)
Rank E / F / G
2 or

:
3 or

or

:

:

or


4 or

or

:

or

:

:

5 or

or

:

or

:

:

or

or


6 or

or

:

or

:

:

or

7 or

or

:

or

:

:

or

or

8 or

or

:

or

:

:

or

or

9 or

or

:

or

:

:

or

or

10 or

or

:

or

:

:

or

11 ... ... ... ...

Read more about this topic:  Dynkin Diagram

Famous quotes containing the word diagrams:

    “Professors could silence me then; they had figures, diagrams, maps, books.... I was learning that books and diagrams can be evil things if they deaden the mind of man and make him blind or cynical before subjection of any kind.”
    —Agnes Smedley (1890–1950)