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.