On semicomplete Lie algebras

Abstract

Let L be a Lie algebra and \(L^2\) be its derived algebra. A derivation \(\alpha \) of L is called \(\mathrm {ID}\)-derivation if \(\alpha (x)\in L^2\) for each \(x\in L\). The set of all \(\mathrm {ID}\)-derivations of L is denoted by \(\mathrm {ID}(L)\). A Lie algebra L is called semicomplete Lie algebra if and only if \(\mathrm {ID}(L)=\mathrm {ad}(L),\) where \(\mathrm {ad}(L)\) is the set of inner derivations of L. In this paper, we give some results on semicomplete Lie algebras and characterization of semicomplete nilpotent Lie algebras of finite dimension.