Strictly Upper Triangular Matrices Over Double Guide And Set Of Nonlinear Three Yuan Lie Algebra On Guide

Let Nn(R)be the algebra consisting of all strictly triangular n×n matrices over a commutative ring R with the identity. An R-bilinear mapφ:Nn (R)×Nn (R)â†'Nn (R) is called a biderivation if it is a derivation with respect to both arguments。If T is a noncommutative algebra, then the map φ(x,y)=λ[x,y],â–½x,y∈T, λ∈Z(T), the center of T is a basic example of biderivation, and we call it an inner biderivation. We will prove that any biderivation of Nn (R) can be decomposed as a sum of an inner biderivation, central biderivation and extremal biderivation for n≥5.Let A be a associative algebra. Define Lie product[a,b]=ab-ba for a,b∈A。A nonlinear mapφ:Aâ†'A is called a nonliner Lie triple derivation, if it satisfys φ([[a,b],c])=[[φ(a),b],c]+[[a,φ(b)],c]+[[a,b],φ(c)]. Let H be a Hilbert space, and N be a nest on H, with TN≠{{0},H}. Letφ:T(N)â†'T(N) be a nonlinear Lie triple derivation onT(N). Then φ(χ)=d(χ)+r(χ)I for x∈(N), where d is an additive derivation of T(N) and Ï„:T(N)â†'IF vanishing at Lie triple products[[a,b],c].