三角代数上Lie三重导子的刻画

Characterizations of Lie Triple Derivations on Triangular Algebras

设u=Tri(A,M,B)是三角代数.证明了在一般的假设下,如果线性映射δ:u→u,满足对任意的U,V,W∈u且UV=UW=0(或U·V=U·W=0),有δ([[U,V],W])=[[δ(U),V],W]+[[U,δ(V)],W]+[[U,V],δ(W)],则对任意U∈u,δ(U)=φ(U)+h(U),其中φ:u→u是一个导子,线性映射h:u→Z(u),满足对任意的U,V,W∈u且UV=UW=0(或U·V=U·W=0),有h([[U,V],W])=0.

Let u=Tri(A,M,B) be a triangular algebra. In this paper,under mild assumptions,we prove that if δ:u→u is a linear map satisfying δ([[U,V],W]) = [[δ(U),V],W] + [[U,δ(V),W] + [[U,V],δ(W)],for any U,V,W ∈u with UV=UW=0(resp. U·V=V·W=0), then δ(U) =φ(U)+h(U) for any U∈u where φ :U→u is a derivation,h:u→Z(u)is a linear map vanishing at second commutators with UV=UW=0(resp.U·V =U·W=0).