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Characterizations of Lie derivations of triangular algebras. (English) Zbl 1226.16026

A triangular algebra T=T(A,X,B) has the form of an upper triangular matrix ring with elements having diagonal entries in A and B and upper right entries in X; A and B are unital algebras over a commutative ring R with 1, and X is an A-B-bimodule that is faithful on each side. The center of T is Z(T)={diag(a,b)Tam=mb for all mX}. The projection of Z(T) to either A or B is central.

The main results in the paper assume that these projections are all of Z(A) and Z(B). The first theorem considers an R-linear map δ of T satisfying δ([x,y])=[δ(x),y]+[x,δ(y)] for all x,yT so that xy=0 and proves that on T, δ(x)=d(x)+τ(x) for d a derivation of T and τ an R-linear map from T to Z(T) with τ([x,y])=0 when xy=0. The second main result is the first with the assumptions that xy=0 replaced with xy=diag(1 A ,0 B ), but also requires the additional assumption that for any aA there is an integer j so that j1 A -a is invertible.


MSC:
16W25Derivations, actions of Lie algebras (associative rings and algebras)
16S50Endomorphism rings: matrix rings
47B47Commutators, derivations, elementary operators, etc.
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