Non-global nonlinear Lie n-derivations on unital algebras with idempotents

Let X be a unital algebra with nontrivial idempotents. For any s1, s2, . . . , sn E X, define p1(s1) = s1, p2(s1, s2) = [s1, s2] and pn(s1, s2, . . . , sn) = [pn-1(s1, s2, . . . , sn-1), sn] for all integers n >= 3. In the present article, it is shown that if a map phi : X -> X satisfies phi(pn(s1, s2, ... , sn)) = Xn i=1 pn(s1, ... ,si-1,phi(si),si+1,... , sn) (n >= 3) for all s1, s2, . . . , sn E X with s1s2 center dot center dot center dot sn = 0, then phi(s + t) - phi(s) - phi(t) E Z(X) for all s, t E X, and under some mild assumptions phi is of the form delta + tau, where delta : X -> X is an additive derivation and tau : X -> Z(X) is a map such that tau(pn(s1, s2, ..., sn)) = 0 for all s1, s2, . . . , sn E X with s1s2 center dot center dot center dot sn = 0. The above results are then applied to certain special classes of unital algebras, namely triangular algebras, full matrix algebras and algebra of all bounded linear operators.