A note on outer derivations of Leibniz algebras

Abstract

In this article, we discuss completeness of non-Lie Leibniz algebras by studying various conditions under which they admit outer derivations. Our study focusses particularly on the class of non-perfect Leibniz algebras whose center is not contained in the Leibniz kernel. We extend to this class of Leibniz algebras several well-known results on derivations of Lie algebras. In particular, we show that solvable Leibniz algebras in this class are not complete. Also, we show that non-stem Leibniz algebras admit outer central derivations. Finally, we independently show that semisimple non-Lie Leibniz algebras are complete.