Research On Several Classes Of Mappings On Triangular Algebras

In this dissertation,by using the structure properties of algebras and the method of decompositing algebras,we mainly study the structure of some mappings on triangular algebras.These mappings include nonlinear generalized Lie derivations,zero point ?-Lie weak derivable and zero point ?-Lie higher weak derivable mappings,Lie invariant mappings and nonlinear(m,n)-Lie centralizers,nonlinear(m,n)-derivable and nonlinear(m,n)-higher derivable mappings.The main body of the present thesis is organized by four chapters as follows:In the first chapter,we introduce the significance and background of the selected topic,recall the present situation and achievements,and offer necessary preliminary concepts and conclusions for later chapters.In the second chapter,we investigate nonlinear generalized Lie derivations on triangular algebras,and prove that every nonlinear generalized Lie derivation on triangular algebras is the sum of an additive generalized derivation and a center mapping vanishing on all commutators.Moreover,we give the general forms of the zero point ?-Lie weak derivable and zero point ?-Lie higher weak derivable mappings on triangular algebras.In the third chapter,we investigate linear mappings on triangular algebras for which the space of all inner derivations is Lie invariant,and prove that such a linear mapping is the sum of a Lie derivation and the identity mapping multiplied by a center element.Meanwhile,we characterize the nonlinear(m,n)-Lie centralizers on|(m-n)(m + n)|-torsion free triangular algebras.In the last chapter,we investigate nonlinear(m,n)-derivable and nonlinear(m,n)-higher derivable mappings,and prove that every nonlinear(m,n)-derivable mapping and every nonlinear(m,n)-higher derivable mapping on |m + n|-torsion free triangular algebras are a derivation and a higher derivation,respectively.In the following,we present the specific results of this thesis.(1)Let u be a triangular algebra with ?A(Z(u))Z(A)and ?B(Z(u))=Z(B).If ? is a nonlinear generalized Lie derivation from u into itself with an as-sociated nonlinear mapping f,then there exist two additive generalized derivations 0 and g on u,respectively,and a mapping ? from u into Z(u)vanishing on all commutators such that ?(x)= ?(x)+ ?(x)and f(x)= g(x)+ ?(x)for all x x ?u.(2)Let u be a triangular algebra over a number field F.If d is a zero point ?-Lie(? ? 1)weak derivable mapping from u into itself,then there exist a derivation ?on u and a center element ? such that d(x)=?(x)+ ?x for all x ?u.(3)Let u be a triangular algebra over a number field F.If D = {dk}k?N is a zero point ?-Lie(?? 1)higher weak derivable mapping from u into itself with dk(1)=0((?)k?N+),then D is a higher derivation.(4)Let u be a triangular algebra with ?A(Z(u))= Z(A)and ?B(Z(u))=Z(B),? be a R-linear mapping from u into itself.If I D(u)is a Lie invariant subspace for ?,then there exist a Lie derivation ? on u and a center element ? such that ?(x)= ?(x)+ ?x for all x? u.(5)Let m,n be fixed integers with(m+n)(m-n)? 0,u be a |(m+n)(m-n)|-torsion free triangular algebra with(?)A(Z(u))= Z(A)and ?B(Z(u))=Z(B).If L is a nonlinear(m,n)-Lie centralizer from u into itself,then there exist a center element A and a mapping ? from u into Z(u)vanishing on all commutators such that L(x)=?x+?(x)for all x ?u.(6)Let m,n be fixed integers with m + n ? 0,u be a |m + n|-torsion free triangular algebra.If d is a nonlinear(m,n)-derivable mapping from u into itself,then d is a derivation.(7)Let m,n be fixed integers with m + n ? 0,u be a |m + n|-torsion free triangular algebra.If D ={dk}k?N is a nonlinear(m,n)-higher derivable mapping from u into itself,then D is a higher derivation.