TY - JOUR
T1 - Constructing simply laced Lie algebras from extremal elements
AU - Draisma, J.
AU - Panhuis, in 't, J.C.H.W.
PY - 2008
Y1 - 2008
N2 - For any finite graph G and any field K of characteristic unequal to 2, we construct 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 G 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 G 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 outside the open dense subset. Our results may prove useful for recognizing these Lie algebras.
AB - For any finite graph G and any field K of characteristic unequal to 2, we construct 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 G 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 G 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 outside the open dense subset. Our results may prove useful for recognizing these Lie algebras.
U2 - 10.2140/ant.2008.2.551
DO - 10.2140/ant.2008.2.551
M3 - Article
SN - 1937-0652
VL - 2
SP - 551
EP - 572
JO - Algebra & Number Theory
JF - Algebra & Number Theory
IS - 5
ER -