三角代数上互逆元处的高阶ξ-Lie可导映射

Higher ξ-Lie derivable maps on triangular algebras at reciprocal elements

设U=Tri(A,M,B)是含单位元1的三角代数,1A、1B分别是A和B的单位元。对任意的A∈A,B∈B分别存在整数k1、k2,使得k11A-A,k21B-B在三角代数中可逆。利用代数分解的方法,证明了如果{φn}n∈N:U→U是一列线性映射满足对任意的U,V∈U且UV=VU=1,有φn([U,V]ξ)=Σi+j=nφi(U)φj(V)-ξφi(V)φj(U)(ξ≠0,1),则{φn}n∈N是U上的高阶导子,其中φ0=id0是恒等映射,[U,V]ξ=UV-ξVU。

Let U=Tri(A,M,B)be a triangular algebra with identity 1,1A,1B be the unit of A and B,respectively.For any A∈A,B∈B,there are integers k1,k2 respectively,making k11A-A,k21B-B invertible in triangular algebras.{φn}n∈N:U→U be a sequence of linear maps.In this paper,we prove that if{φn}n∈N satisfiesφn([U,V]ξ)=Σi+j=nφi(U)φj(V)-ξφi(V)φj(U)(ξ≠0,1),for any U,V∈U with UV=VU=1,then{φn}n∈N is a higher derivation,whereφ0=id0 is the identity map,[U,V]ξ=UV-ξVU.