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

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)-biomodule. Assume that xi is an element of 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 is an element of U with [X, Y] = XY - YX = 0 if and only if L(X) = phi(X) + ZX + f(X) for all Lambda, where phi is a linear derivation, Z is a central element and f is a central valued linear map. For the case 1 not equal xi is an element of F, L is additive and satisfied L([X, Y](xi)) = [L(X), Y](xi) + [X, L(Y)](xi) for any X, Y is an element of U with [X, Y](xi) = XY - xi Y X = 0 is and only if L(I) is in the center of U and L(A) = phi(A) + L(I)A for all A, where phi is an additive derivation satisfying phi(xi A) = xi phi(A) for each A. In addition, all additive maps L satisfying L([X, Y](xi)) = [L(X), Y](xi) +[X, L(Y)](xi) for any X, Y is an element of U with XY = 0 are also characterized.