Jan Draisma and Jos in ’t panhuis ALGEBRA AND NUMBER THEORY 2:5(2008)

Constructing simply laced Lie algebras from extremal elements Jan Draisma and Jos in ’t panhuis For any finite graph 0 and any field K of characteristic unequal to 2, we con-struct an algebraic variety X over K whose K-points parametrize K-Lie algebras generated by extremal elements, corresponding to the vertices of the graph, with prescribed commutation relations, corresponding to the nonedges. After that, we study the case where 0 is a connected, simply laced Dynkin diagram of finite or affine type. We prove that X is then an affine space, and that all points in an open dense subset of X parametrize Lie algebras isomorphic to a single fixed Lie algebra. If 0 is of affine type, then this fixed Lie algebra is the split finite-dimensional simple Lie algebra corresponding to the associated finite-type Dynkin diagram. This gives a new construction of these Lie algebras, in which they come together with interesting degenerations, corresponding to points out-side the open dense subset. Our results may prove useful for recognizing these Lie algebras. 1. Introduction an