Images of inner derivations of free Lie algebras and superalgebras (Q1345674)

Let \(K\) be an associative commutative ring with unit and let \(L(X)\) be the free Lie \(K\)-algebra canonically embedded into the free associative algebra \(K\langle x\rangle\). By a result of \textit{D. Z. Đoković} [J. Algebra 119, 233-245 (1988; Zbl 0757.17004)] the image \(\text{Im} (\text{ad } x)\), \(x\in X\), of the adjoint operator \(\text{ad } x: v\to [x,v]\), \(v\in L(X)\), is a free \(K\)-submodule of \(L(X)\) and \(\text{Im} (\text{ad } x)\) is a direct summand of the \(K\)-module \(L(X)\). The purpose of the paper under review is to extend this result to the case of \(\text{Im} (\text{ad } u)\), where the coefficient of the leading term of \(u\in L(X)\) (\(u\) is considered as an element of \(K\langle X\rangle\)) is invertible in \(K\). The author also establishes an analogue of this result for chromatic Lie superalgebras and restricted chromatic Lie superalgebras under some natural additional restrictions on \(u\).