On Lie Algebras Generated by Few Extremal Elements
Dan Roozemond

An element x of a Lie algebra L is called extremal if [x,[x,y]] is a scalar 
multiple of x for all y in L. If L is a Lie algebra of Chevalley type, then the 
extremal elements are precisely the long root elements, so they are quite a 
natural object to study. We consider the setup where a Lie algebra L is 
generated by a small number of extremal elements whose commutation relations 
are prescribed by a given graph. In particular, for a  graph $\Gamma$ and a field K we 
consider an algebraic variety X over K whose K-points parametrize Lie algebras 
generated by extremal elements. 

I'll try to make this talk quite accessible by discussing some basic 
previously-known properties of such Lie algebras, in the process hopefully 
drawing some pretty pictures on the blackboard, and I'll describe some new 
results obtained by computer calculations in Magma. These results suggest a 
connection between graph planarity of $\Gamma$ and the structure of X, hoping to 
attract members of both the CO group and the DAM group.