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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.

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