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".

Here is the
Dynkin diagram.
Click on any node to edit its label.

Click this box to show a test dynamic 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.

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:

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.

You will get a "Save As..." dialog. Skip to the next step.

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{}.

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 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.]

Copyright (c) 2006-2007 Alexey Spiridonov

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.