Lie Derivations on Generalized Matrix Algebras by Local Actions

Let G=G(A,B,M,N) be a generalized matrix algebra. A linear map Δ:G→G is called a Lie derivation at E∈G if Δ([U,V])=[Δ(U),V]+[U,Δ(V)] for all pairs U,V∈G such that UV=E. In this paper, we use techniques of matrix decomposition and algebraic identity analysis to fully characterize the general form of Lie derivations at E=e0000, where e0 is an arbitrary fixed element in A. Our main result establishes a necessary and sufficient condition for a Lie derivation at E=e0000 to be decomposable into the sum of a derivation of G and a center-valued linear map. This characterization significantly extends the classical results concerning global Lie derivations and provides a deeper insight into the local Lie-type behavior in operator algebras.