Non-additive Lie centralizer of strictly upper triangular matrices

Abstract

Let ℱ be a field of zero characteristic, let Nn(ℱ) denote the algebra of n X n strictly upper triangular matrices with entries in ℱ, and let ƒ : Nₙ(ℱ) → Nₙ(ℱ) be a non-additive Lie centralizer of Nₙ(ℱ), that is, a map satisfying that ƒ([X; Y ]) = [ƒ(X); Y ] for all X; Y ∈ Nₙ(ℱ). We prove that ƒ(X) = ⋋ X + ƞ (X) where ⋋ ∈ ℱ and ƞ is a map from Nₙ(ℱ) into its center Ƶ (Nₙ(F)) satisfying that ƞ([X; Y ]) = 0 for every X; Y in Nₙ(ℱ).