Graphs for Classical Lie Algebras

A nonzero element x of a Lie algebra L over a field F is called extremal if [x, [x,L]] ⊆ Fx. Extremal elements are a well-studied class of elements in simple finite-dimensional Lie algebras of Chevalley type: they are the long root elements. In [CSUW01], Cohen, Steinbach, Ushirobira and Wales have studied Lie algebras generated by extremal elements, in particular those of Chevalley type. The authors also find the minimum size of a set of generating extremal elements for the Lie algebras of Chevalley type and find such minimal generating sets of extremal elements explicitly. In the present paper, we also find such minimal generating sets of extremal elements explicitly for the four classical families of Lie algebras: those of type An, Bn, Cn and Dn. We will do this in a more geometrical setting and will find criteria for sets of extremal elements to generate Lie algebras of this type. By Lemma 2.2, each Lie algebra generated by a pair of linearly independent extremal elements is in one of only three isomorphism classes: either the two-dimensional commutative Lie algebra, or the so-called Heisenberg Lie algebra h, or sl2. Given a generating set S of extremal elements, we examine the subalgebras generated by pairs of these elements. These give rise to graphs: the vertices correspond to the elements of S, and two vertices are adjacent if the corresponding extremal elements generate a three-dimensional algebra and nonadjacent if they commute. We will say that the Lie algebra generated by S realizes this graph. We find one such graph for each Lie algebra of classical Chevalley type, depicted in Figures 1.1 up to 1.4, and show that if a Lie algebra realizes one of these graphs, then in the generic case it is isomorphic to the Lie algebra of the corresponding Chevalley type, in the following sense. Given a graph Γ, we define a vector space V(Γ) parametrizing the Lie algebras that realize Γ. Let X(Γ) be the subset of V(Γ) of values f for which the associated Lie algebra L(Γ, f) has maximal dimension among such algebras for the same graph Γ. We will see in Lemma 3.5 that X(Γ) carries the structure of an affine variety. The following theorem and analogues for the three other families of Chevalley type Lie algebras will be our main results.