On \(\delta\)-derivations of Lie algebras (Q1972205)

A linear mapping \(\varphi\) of an algebra \(A\) over a ring of scalars \(\Phi\) is called a \(\delta\)-derivation of \(A\), for a certain \(\delta\in\Phi\), if the equality \((xy)^{\varphi}=\delta(x^{\varphi}y+xy^{\varphi})\) holds for all \(x,y\in A\). The 1-derivations of \(A\) are exactly the ordinary derivations of \(A\); the elements of the centroid \(\Gamma (A)\) provide examples of 1/2-derivations; the 0-derivations of \(A\) can be described as the linear mappings with zero restrictions to \(A^2\). The \(\delta\)-derivations of these types are called trivial. The author first notices that every \(\delta\)-derivation of a Lie algebra \(L\) (with \(\delta\neq 0,1\)) provides a certain \(R\)-matrix for \(L\). The main result of the article states that a prime Lie \(\Phi\)-algebra \(L\) of characteristic different from \(2,3\), with a nondegenerate symmetric invariant bilinear form, has no nontrivial \(\delta\)-derivations, except the case in which \(\delta=-1\) and \(L\) is a central order in a simple 3-dimensional Lie algebra over a field.