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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.
##### Keywords:
Lie derivations; triangular algebras; linear maps; commutators
Full Text:
##### References:
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