A geometric characterization of the classical Lie algebras

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.…”

“…The extremal form ĝ on ĝ satisfies ĝ(x ⊗ λ, y ⊗ µ) = g(x, y)λµ for all x, y ∈ E and λ, µ ∈ K. Now we can formulate the main result from [5]:…”

“…If h is alternating (i.e., σ is the identity), the algebra s h (V, V * ) is isomorphic to the finitary symplectic Lie algebra fsp(V, h) := x ∈ sl(V, V * ) | h(x(v), w) + h(v, x(w)) = 0 for all v, w ∈ V }, see [5], while for proper skew-Hermitian forms (i.e., σ has order 2), the Lie algebra s h (V, V * ) over…”

“…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.…”

“…We will also consider several other subalgebras of fsl(V, V * ) generated by infinitesimal transvections. If V is equipped with a non-degenerate symplectic form f , then the infinitesimal transvections t v,φ with 0 = v ∈ V and φ ∈ V * given by φ(w) = f (v, w) are contained in and generate the finitary symplectic Lie algebra fsp(V, f ), see [5]. The extremal geometry of this Lie algebra consist then of all point-hyperplane pairs (p, H), where H is the hyperplane of all points perpendicular to p with respect to f .…”