Lie algebras generated by extremal elements

August 31, 2005

This project is finished now! On August 17 I had my final presentation and discussion, and I received a 9 for this project, which was rather satisfying. Yesterday, I received my Master's degree.
The report can be found here.

June 15, 2005

So it's been a while since I wrote something here. That may have been partly caused by the fact that the program I wrote didn't give the desired results. Though that may have been caused partly by the implementation, I have now come to think that my expectations may have been a bit too high as well.

After all, I decided to shift my attention to something else. After messing around with various large expressions I have found a few proofs. Firstly, I proved that certain degenerate cases of a Lie algebra L generated by extremal elements ('degenerate' means that certain elements commute) are isomorphic to A_n, and other degenerate cases are isomorphic to C_n.
Furthermore, I found a proof for the following theorem: If L₁ is generated by n extremal elements, and L₂ is generated by m extremal elements, then the direct sum L₁ + L₂ cannot be generated by less than n+m extremal elements.

If you happen to be interested in the proofs, you can send an e-mail to .

February 3, 2005

In the past two weeks I have reproduced some of the results from the 99 paper, including that the most generic one generated by three extremal elements is isomorphic to sl₃, aka A2. Studying the four extremal elements case showed that the most general case is indeed isomorphic to D4 (!), but I also found several occurences of A2, A3, B3, G2, B2, C2, A1, and A1xA1. More information will follow!

January 20, 2005

So I have been working on this project for a while now, and I should tell something about the progress here. I now more or less know what a Lie Algebra is and how to work with extremal elements. Moreover, I have been working on Lie algebras generated by four extremal elements, as the paper by Cohen et al. below does not fully cover that case.

In the paper it is proved that the most generic Lie algebra generated by two extremal elements is isomorphic to sl₂, and the most generic one generated by three extremal elements is isomorphic to sl₃. Now you must think that the most generic one generated by four extremal elements (which I'll refer to as L) is isomorphic to sl₄, however the dimension of L is 28, and the dimension of sl₄ is 15, so they can never be isomorphic. The only simple Lie algebra of dimension 28 is D₄, so I'm now looking into that possibility.

November 30, 2004

An interesting paper on the subject: Paper by Cohen, Steinbach, Ushirobira, Wales. It is a paper called Lie Algebras Generated by Extremal Elements. This is the topic my Master's project is on, and a significant part of it is an extension to this paper.