Lie triple derivations of triangular algebras. (Q442685)

Let \(R\) be a 2-torsion free commutative ring with identity element, \(A\) and \(B\) unital \(R\)-algebras, and \(M\) an \(A\)-\(B\) bi-module that is faithful on each side. These define a formal triangular algebra \(T(A,M,B)=T=A\oplus M\oplus B\) as an additive group, with \((a,m,b)\cdot(a',m',b')=(aa',am'+mb',bb')\) the multiplication in \(T\). Note that the center of \(T\), \(Z(T)=\{(a,0,b)\in T\mid a\in Z(A),\;b\in Z(B)\), and \(am=mb\) for all \(m\in M\}\).NEWLINENEWLINE A Lie triple derivation of any ring \(W\) is an additive \(D\colon W\to W\) so that for all \(x,y,z\in W\), \(D([[x,y],z])=[[D(x),y],z]+[[x,D(y)],z]+[[x,y],D(z)]\).NEWLINENEWLINE The main result of the authors shows that if \(D\) is a Lie triple derivation of \(T\), as above, if the projections of \(Z(T)\) to \(Z(A)\) and \(Z(B)\) are surjective, and if either \(Z(A)=\{a\in A\mid [[a,x],y]=0\) for all \(x,y\in A\}\) or \(Z(B)=\{b\in B\mid [[b,x],y]=0\) for all \(x,y\in B\}\) then \(D=d+f\) for \(d\in\text{Der}(T)\) and \(f\colon T\to Z(T)\), an \(R\)-module map so that \(f([[T,T],T])=0\). One application of this result applies to any Lie triple derivation of the usual upper triangular matrix algebra \(T_n(R)\). In this case, since \(R\) is commutative, the additional hypotheses in the main theorem hold automatically, and it also happens that \(f([T,T])=0\).