A note on derivations of simple algebras
A derivation of a nonassociative algebra \(A\) is called inner if it is contained in the Lie multiplication algebra \({\mathcal L}(A)\) of \(A\). For several classes of algebras it is known that the simple algebras in the respective class have only inner derivations. On the other hand, there are simple algebras (of finite dimension over an algebraically closed field of zero characteristic) which have non-inner derivations, as shown by the reviewer. Also, for simple finite dimensional algebras over an algebraically closed base field of characteristic zero there exist non- inner derivations only if every left or right multiplication by an element of \(A\) has zero trace. (In that case, derivations with nonzero trace, if they exist, are not inner.) This follows from a theorem of Sagle and Walde [\textit{A. A. Sagle} and \textit{R. E. Walde}, Introduction to Lie groups and Lie algebras (Academic Press, New York 1973; Zbl 0252.22001)]. In this paper, the authors extend the results above to arbitrary base fields of characteristic zero. (Note that derivations behave nicely under base field extensions in the characteristic zero case, due to separability!) Furthermore, they discuss the relation between semisimplicity of \({\mathcal L}(A)\) and innerness of derivations, and show that the existence of a nondegenerate trace form forces all derivations to be inner.
scientific article; zbMATH DE number 589965