Multiplicative Lie \(n\)-derivations of generalized matrix algebras.
A nonlinear Lie \(n\)-derivation of an algebra \(A\) is a (not necessarily linear) map of \(A\) that acts as a derivation on the polynomial \([[\cdots[x_1,x_2],\dots],x_n]\). The main result states that a nonlinear Lie \(n\)-derivation of a generalized matrix algebra is, under certain technical assumptions, the sum of an (additive) derivation and a map with the range in the center that vanishes on commutators. As a corollary it is shown that if \(A\) is a full matrix algebra \(M_n(B)\), where \(B\) is a unital algebra, then this holds under rather mild assumptions. This was conjectured by \textit{D. Benkovič} and \textit{D. Eremita} [Linear Algebra Appl. 436, No. 11, 4223-4240 (2012; Zbl 1247.16040)].
scientific article; zbMATH DE number 6138439
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