On marginal automorphisms of free nilpotent Lie algebras (Q780510)

Let \(G\) be a group, and \(w(x_{1}, \ldots, x_{n})\) be a (group) word. \textit{P. Hall} had defined in [J. Reine Angew. Math. 182, 156--157 (1940; Zbl 0023.29902)] the marginal subgroup of \(G\) with respect to \(w\) as \(\{ g \in G : w(g_{1}, \dots, g_{i} g, \ldots, g_{n}) = w(g_{1}, \dots, g_{i}, \dots, g_{n})\ \mathrm{for}\ \mathrm{all}\ i \ \mathrm{and}\ \mathrm{all}\ g_{j} \in G \}\). For instance the centre of \(G\) is the marginal subgroup with respect to the commutator word \([x_{1}, x_{2}]\). Given a non-empty set \(W\) of words, one can define the marginal subgroup \(W^{ \star}(G)\) as the intersection of all the marginal subgroups of \(w \in W\). \textit{M. Shabani Attar} has defined in [Commun. Algebra 37, No. 7, 2300--2308 (2009; Zbl 1177.20040)] an automorphism \(\alpha\) of \(G\) to be marginal with respect to such a set \(W\) if \(g^{-1} \alpha(g) \in W^{\star}(G)\) for all \(g \in G\). For instance when one takes \(W\) to consists of the single commutator word, one obtains the central automorphisms. In the paper under review, the author extends and explores these definitions in the context of Lie algebras.