CHARACTERIZING LIE (ξ-LIE) DERIVATIONS ON TRIANGULAR ALGEBRAS BY LOCAL ACTIONS ∗

Abstract. Let U = Tri(A,M,B) be a triangular algebra, where A, B are unital algebras over a field F and M is a faithful (A,B)-bimodule. Assume that ξ ∈ F and L: U → U is a map. It is shown that, under some mild conditions, L is linear and satisfies L([X,Y]) = [L(X),Y]+[X,L(Y)] for any X,Y ∈ U with [X,Y] = XY −YX = 0 if and only if L(X) = ϕ(X)+ZX +f(X) for all A, where ϕ is a linear derivation, Z is a central element and f is a central valued linear map. For the case 1 ̸ = ξ ∈ F, L is additive and satisfies L([X,Y]ξ) = [L(X),Y]ξ + [X,L(Y)]ξ for any X,Y ∈ U with [X,Y]ξ = XY −ξYX = 0 if and only if L(I) is in the center of U and L(A) = ϕ(A) +L(I)A for all A, where ϕ is an additive derivation satisfying ϕ(ξA) = ξϕ(A) for each A. In addition, all additive maps L satisfying L([X,Y]ξ) = [L(X),Y]ξ +[X,L(Y)]ξ for any X,Y ∈ U with XY = 0 are also characterized