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  • Nilpotent subspaces of maximal dimension in semi-simple Lie algebras

  • Jan Draisma, Hanspeter Kraft, Jochen Kuttler
  • Journal:
    Compositio Mathematica / Volume 142 / Issue 2 / March 2006
    Published online by Cambridge University Press:
    13 March 2006, pp. 464-476
    Print publication:
    March 2006
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  • We show that a linear subspace of a reductive Lie algebra $\operatorname{\mathfrak g}$ that consists of nilpotent elements has dimension at most $\frac{1}{2}(\dim\operatorname{\mathfrak g}-\operatorname{rk}\operatorname{\mathfrak g})$, 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 $(n\times n)$-matrices.