Nilpotent subspaces of maximal dimension in semi-simple Lie algebras

[摘要] We show that a linear subspace of a reductive Lie algebra $operatorname{mathfrak g}$ that consists of nilpotent elements has dimension at most $frac{1}{2}(dimoperatorname{mathfrak g}-operatorname{rk}operatorname{mathfrak g})$, and that any nilpotent subspace attaining this upper bound is equal to the nilradical of a Borel subalgebra of $operatorname{mathfrak g}$. This generalizes a classical theorem of Gerstenhaber, which states this fact for the algebra of $(nimes n)$-matrices.

[学科分类] 数学（综合）