Recovering the Lie algebra from its extremal geometry

Abstract

An element x of a Lie algebra L over the field F is extremal if [x, [x, L]] = F x. Under minor assumptions, it is known that, for a simple Lie algebra L, the extremal geometry E(L) is a subspace of the projective geometry of L and either has no lines or is the root shadow space of an irreducible spherical building ∆. We prove that if ∆ is of simply-laced type, then L is a quotient of a Chevalley algebra of the same type.

A geometric characterization of the finitary special linear and unitary Lie algebras

“…This result together with the following theorem, which combines the results of [6] and [8], provides us with a characterization of most of the classical Lie algebras.…”

“…x ji ] form a Chevalley spanning set as described in Section 4 of [8]. But then, using the group G = Exp(x) | x ∈ E 0 it readily follows by arguments as in Section 6 and 7 of [8] that the Lie algebra generated by this Chevalley spanning set contains E 0 and hence equals g. But then g is a Chevalley Lie algebra of type A n−1 and hence isomorphic to (p)sl(V ).…”

“…In this section we prove Theorem 1.1 and show that a simple Lie algebra over a field F generated by its set of pure extremal elements and its extremal geometry isomorphic to Γ(V, Π), where V is an F-vector space of dimension at least 3 and Π a subspace of V * with Ann V (Π) = {0}, is isomorphic to (p)fsl(V, Π). If V is finite dimensional, then the theorem follows from [8] ( see also [10]). Indeed, using [8] we obtain the following theorem.…”

“…If V is finite dimensional, then the theorem follows from [8] ( see also [10]). Indeed, using [8] we obtain the following theorem. (Here we say that a subset E 0 of E is closed under the action of a subgroup G of Aut(g) if and only if x g ∈ E 0 for all x ∈ E 0 and g ∈ G.) Theorem 4.1 (Cuypers, Roberts, and Shpectorov).…”

“…The main result of Cohen and Ivanyos [2,3] states that for a finite dimensional simple Lie algebra g generated by its pure extremal elements the extremal geometry Γ(g) either contains no lines or is a so-called root shadow space of a spherical building. Subsequently, in combined work of Fleischmann, Roberts, Shpectorov and the first author of this paper [8,6], it has been shown that in case Γ(g) is a root shadow space of a spherical building of rank at least 3, the isomorphism type of g is uniquely determined by its extremal geometry. In particular, classical Lie algebras with an extremal geometry of finite rank at least 3 are characterized by this extremal geometry.…”