Strong commutativity preserving maps on rank k strictly triangular matrices

Abstract

Let n ⩾ 2 be an integer and let N n ( F ) be the algebra of n × n strictly upper triangular matrices over a field F with center Z ( N n ( F ) ) . In this paper, we obtain a structural characterization of additive maps ψ : N n ( F ) → N n ( F ) satisfying [ ψ ( A ) , ψ ( B ) ] − [ A , B ] ∈ Z ( N n ( F ) ) for all A , B ∈ N n ( F ) . We deduce from this result a characterization of strong commutativity preserving additive maps on rank k strictly upper triangular matrices over F , where 1 ⩽ k ⩽ n − 1 is a fixed integer such that k ≠ n − 1 when F is the Galois field of two elements.