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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 \(Z(T)=\{\text{diag}(a,b)\in T\mid am=mb\) for all \(m\in X\}\). 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 \(\delta\) of \(T\) satisfying \(\delta([x,y])=[\delta(x),y]+[x,\delta(y)]\) for all \(x,y\in T\) so that \(xy=0\) and proves that on \(T\), \(\delta(x)=d(x)+\tau(x)\) for \(d\) a derivation of \(T\) and \(\tau\) an \(R\)-linear map from \(T\) to \(Z(T)\) with \(\tau([x,y])=0\) when \(xy=0\). The second main result is the first with the assumptions that \(xy=0\) replaced with \(xy=\text{diag}(1_A,0_B)\), but also requires the additional assumption that for any \(a\in A\) there is an integer \(j\) so that \(j1_A-a\) is invertible.

MSC:
16W25 Derivations, actions of Lie algebras
16S50 Endomorphism rings; matrix rings
47B47 Commutators, derivations, elementary operators, etc.
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