Groups, Algebras and Algorithms

For a finite graph Gamma we study Lie algebras with a distinguished set of generators corresponding to the nodes of Gamma which commute when the corresponding nodes in Gamma are not connected. By results of Kostrikin-Zelmanov and Cohen-Steinbach-Ushirobira-Wales there is a uniform bound on the dimension of such Lie algebras depending only on Gamma. We show that the "moduli space" of those Lie algebras (with distinguished generators) that attain the maximal possible dimension carries a natural structure of a finite-dimensional affine variety. For particular graphs Gamma, this variety has been found to be an affine space and the existence of a "generic" Lie algebra has been proved. I will review work by in 't panhuis, Postma, Roozemond, and myself to that effect.