Nonlinear Lie Derivations Of Generalized Matrix Algebras

In2001,Cheung studied the problem on commuting mappings of triangularalgebras,which created a precedent for the research of mappings of triangularalgebras.From then on,the research of mappings of triangular algebras have costantlyproduced,making some important problems of mappings of triangular algebrasresolved.In broad sense,generalized matrix algebras include triangular algebras andfull matrix algebras over a unital algebra,which are developed from the concept oftriangular algebras.In2010,Xiao and Wei firstly extended commuting mappings oftriangular algebras to generalized matrix algebras.In recent years,people have startedto extended results of mappings of triangular algebras to generalized matrixalgebras.Now there are only few research results of generalized matrix algebras,andmany important mappings problems have not been disscussed so far.In this paper,our main aim is to investigate nonlinear Lie derivations ofgeneralized matrix algebras.We will give a sufficient condition for every nonlinear Liederivations of a certain class of generalized matrix algebras to be of standardform.Our main results completely generalize a result on nonliear Lie derivations oftriangular algebras.As a consequence,nonliear Lie derivations of full matrix algebraover a unital algebra are determined.