Lie triple derivations of triangular algebras

Let R be a commutative ring with identity, A,B be unital algebras over R and M be a unital (A,B)-bimodule, which is faithful as a left A-module and also as a right B-module. Let be the triangular algebra consisting of A,B and M. This work is motivated by some intensive works of Brešar [4], , Cheung [9], and Zhang et al. [30]. Here, we study Lie triple derivations of T. It is shown that under mild assumptions, every Lie triple derivation on T is of standard form. That is, L can be expressed through an additive derivation and a linear functional vanishing on all second commutators of T. Examples of Lie triple derivations on some classical triangular algebras are supplied.