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 × n strictly upper triangular matrices with entries in F , and let f : Nn (F ) → Nn (F ) be a non-additive Lie centralizer of Nn (F ) , that is, a map satisfying that f ([X, Y ]) = [f (X), Y ] for all X, Y ∈ Nn (F ) . We prove that f (X) = λX + η (X) where λ ∈ F and η is a map from Nn (F ) into its center Z (Nn (F ) ) satisfying that η([X, Y ]) = 0 for every X, Y in Nn (F ) .

Keywords

• Lie centralizer
• strictly upper triangular matrices
• commuting map