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Nonlinear Lie derivations of triangular algebras. (English) Zbl 1193.16030

Let $𝒜$ be an algebra over a commutative ring $ℛ$. A map $\delta :𝒜\to 𝒜$ is called an additive derivation if it is additive and satisfies $\delta \left(xy\right)=\delta \left(x\right)y+x\delta \left(y\right)$ for all $x,y\in 𝒜$. If there exists an element $a\in 𝒜$ such that $\delta \left(x\right)=\left[x,a\right]$ for all $x\in 𝒜$, where $\left[x,a\right]=xa-ax$ is the Lie product or the commutator of the elements $x,a\in 𝒜$, then $\delta$ is said to be an inner derivation. Let $\varphi :𝒜\to 𝒜$ be a map (without the additivity assumption). We say that $\varphi$ is a nonlinear Lie derivation if $\varphi \left(\left[x,y\right]\right)=\left[\varphi \left(x\right),y\right]+\left[x,\varphi \left(y\right)\right]$ for all $x,y\in 𝒜$.

The structure of additive or linear Lie derivations on matrix algebras has been studied by many authors. W.-S. Cheung [in Linear Multilinear Algebra 51, No. 3, 299-310 (2003; Zbl 1060.16033)] initiated the study of Lie derivations of triangular algebras and showed that every Lie derivation of a triangular algebra is of standard form. Recently, L. Chen, J.-H. Zhang, [Linear Multilinear Algebra 56, No. 6, 725-730 (2008; Zbl 1166.16016)], described nonlinear Lie derivations of upper triangular matrix algebras.

Motivated by the previous works, the authors prove that under mild conditions any nonlinear Lie derivation of triangular algebras is the sum of an additive derivation and a map into its center sending commutators to zero. This result is applied to some special triangular algebras, for example to block upper triangular matrix algebras and to nest algebras.

##### MSC:
 16W25 Derivations, actions of Lie algebras (associative rings and algebras) 16S50 Endomorphism rings: matrix rings 16W10 Associative rings with involution, etc. 47B47 Commutators, derivations, elementary operators, etc. 47L35 Nest algebras, CSL algebras