Nilpotent subspaces of maximal dimension in semi-simple Lie algebras

Abstract

We show that a linear subspace of a reductive Lie algebra g that consists of nilpotent elements has dimension at most 1 2 (dim g-rk g), and that any nilpotent subspace attaining this upper bound is equal to the nilradical of a Borel subalgebra of g. This generalizes a classical theorem of Gerstenhaber, which states this fact for the algebra of (nn)-matrices.