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".

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

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.
A. L. Onishchik, E.B. Vinberg, Lie groups and algebraic Groups, Springer, 1990.
V. Kac, Infinite-dimensional Lie algebras, Cambridge University Press, 1990, Chapter 4.
S. Helgason, Differential geometry, Lie groups, and symmetric spaces, American Mathematical Society, 2001, Chapter 10, Section 5.
N. Bourbaki, Lie groups and Lie algebras, Chapters 4-6, Springer, 2002.

Maximum Sizes of Diagrams [hide]

If you need larger diagrams, you can enable them here. However, the program won't look very pretty. Sorry :)

Generate Dynkin diagram series up to size:

Generate affine Dynkin diagram series up to size:

Using This Program Offline [hide]

It's easy to use this program even when you don't have Internet access. Just save this page ("dynkin-diagrams.html") on your computer. This is usually done through "File -> Save As" or "File -> Save Page As". There is one caveat. You need to change the "Save as type" field to say "Web Page, HTML only", not "Web Page, complete" or "Web Archive"; otherwise, the saved file won't work correctly. To use the saved file, open it in your browser.

Inserting Dynkin diagrams into LaTeX papers [hide]

Here are simple, step-by-step instructions for professional-looking Dynkin diagrams in your every paper.

Click "Save diagram as an EPS file" above.
One of three things may happen:
1. You will get a dialog with a choice of "Open" or "Save to disk". Choose "Save to disk", and click OK. If you get a "Save As..." dialog, go to the next step.
  Otherwise, your browser is probably set up to automatically save files to a fixed location (check the Desktop). You are looking for a file with a cryptic name, and a .ps extension. You need to rename and move this file to your LaTeX directory; consult step 4, and skip to step 6.
2. You will get a "Save As..." dialog. Skip to the next step.
3. You get nothing or garbage. Click here in order to save an EPS file by hand.
You should now see a "Save As..." dialog box. Find the directory containing your LaTeX files.
Now, you need to choose a sensible name for the file, like
```
F4-extended.eps
```
Note that the file must have an .eps extension.
Click "Save".
Now, open your LaTeX file. Add
```
\usepackage{graphics}
```
to the preamble. Find the place in the text where you want the diagram. Insert the following code there:
```
\begin{figure}
  \begin{center}
    \includegraphics{}
  \end{center}
  \caption{\label{} 
           This is the Dynkin diagram for $$.}
\end{figure}
```
You might need to adjust the file name; note that the .eps extension is omitted). You can change the size of the picture using the width option:
```
\includegraphics[width=2in]{}
```
You can refer to the figure like this:
```
If you are not convinced, contemplate Figure \ref{}.
```

That's all, folks!

An alternative way of saving the EPS file [hide]

Here are the the contents of the EPS file, in case the "Save as EPS" button doesn't work for you. If your browser saves files automatically, check your download directory first. The file might have a .ps extension.

You can copy and paste the text below into your favorite editor, and save the file with an .eps extension.

The MIT X License [hide]

[The license text comes from here. It is only being applied to the text outside the <script> tags on this page. The JavaScript code is under the GNU GPLv2.]

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

Please write to Alexey Spiridonov, le sha at mit, minus the extra space, and plus edu. 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.