Dynkin Diagrams

The program lets you choose labels for the vertices of Dynkin diagrams. It has a catalog of all Dynkin diagrams (up to size 8), and of all extended and affine Dynkin diagrams (up to size 9). It can be used offline.

Pick a diagram type, and, if needed, check the box "Extended diagram".

Dynkin Diagrams

Extended diagram

Output
Using EPS Diagrams in LATEX
Affine Dynkin Diagrams [more info]
Node Labels [more info]

Number the nodes canonically:

Dynkin ordering [1]
Bourbaki ordering [5]

Options
Reverse arrows
(for Kac-Moody algebras)

What Are Affine Dynkin Diagrams?

Affine Dynkin diagrams, together with extended Dynkin diagrams, completely describe all generalized Cartan matrices of affine type. They are useful in working with Kac-Moody algebras, and in the theory of symmetric spaces (see, e. g., [3] and [4]). The diagrams on this site follow [3].

What Are Labelings?

Various labelings are widely used in the theory of Lie groups and algebras (for instance, to describe weights of linear representations or nilpotent orbits). The box above gives some useful choices.

For an extended or affine diagram, the corresponding Cartan matrix is singular. One of the above options shows a vector from its kernel. This vector is unique (up to scaling), because we can get a standard Dynkin diagram by deleting one node from any affine or extended diagram. See [3] for more information.

The other options present two standard enumerations of the vertices. One was introduced by Dynkin [1] (see also [2]). Another was suggested by Bourbaki [5].

References

  1. E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Americ. Math. Soc. Transl., Series 2, Vol. 6 (1957), 245-378. Reprinted in: Selected Papers of E. B. Dynkin with Commentary, American Mathematical Society, International Press, 2000.
  2. A. L. Onishchik, E.B. Vinberg, Lie groups and algebraic Groups, Springer, 1990.
  3. V. Kac, Infinite-dimensional Lie algebras, Cambridge University Press, 1990, Chapter 4.
  4. S. Helgason, Differential geometry, Lie groups, and symmetric spaces, American Mathematical Society, 2001, Chapter 10, Section 5.
  5. N. Bourbaki, Lie groups and Lie algebras, Chapters 4-6, Springer, 2002.

Please write to Alexey Spiridonov, at , minus the extra space, and plus . Feel free to send questions, suggestions, patches, or whatever else comes along.

Eugene B. Dynkin suggested the idea for this page, helped test it, and contributed many thoughts on how to make it better. Thanks!

I developed this page in Firefox. Users of other browsers -- do let me know how things go for you. Known issues: "Save EPS" doesn't work in IE or Opera; hints on fixing this would be appreciated.

The JavaScript code of this program (the part enclosed between the <script> tags) is licensed under the GNU GPLv2. The remaining text on the page, and the XHTML / CSS markup are licensed under the MIT X License.