Non-additive Lie centralizer of strictly upper triangular matrices
doi:10.17398/2605-5686.34.1.77
Keywords:
Lie centralizer, strictly upper triangular matrices, commuting mapAbstract
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 ) .
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