Nonlinear Lie derivations on upper triangular matrices

Abstract

Let M n (𝔸) and T n (𝔸) be the algebra of all n × n matrices and the algebra of all n × n upper triangular matrices over a commutative unital algebra 𝔸, respectively. In this note we prove that every nonlinear Lie derivation from T n (𝔸) into M n (𝔸) is of the form A → AT − TA + A ϕ + ξ(A)I n , where T ∈ M n (𝔸), ϕ : 𝔸 → 𝔸 is an additive derivation, ξ : T n (𝔸) → 𝔸 is a nonlinear map with ξ(AB − BA) = 0 for all A, B ∈ T n (𝔸) and A ϕ is the image of A under ϕ applied entrywise.