On Lie algebras with only inner derivations

Abstract

It is the purpose of this article to give a characterization of abelian by nilpotent Lie algebras which have only inner derivations. The motivation has been provided by two recent papers [3, 41 which deal with a similar problem in group theory. The Lie algebra case allows some modifications and simplifications as compared to the group theory case. We begin with a brief summary about Lie algebras which have only inner derivations and also about Lie algebras which must possess outer derivations. If the algebra has nondegenerate Killing form, then all derivations are inner [6, p. 741, while if the algebra is nilpotent, then there exist outer derivations [S]. This latter result has been extended in a variety of ways (see [ 131 and the bibliography given there). In particular, we men- tion that if the algebra is solvable and has no outer derivations, then its center must be 0 [2, 121. On the other hand there are examples of Lie algebras with nonzero center and only inner derivations (see [9]). Also a general method for constructing algebras with only inner derivations is described in [7]. In another direction, Schenkman’s famous derivation tower theorem [lo] asserts that an algebra with zero center can be sub- invariantly embedded in an algebra with only inner derivations. Schenkman’s paper also provides bounds on this embedding which are actually met as is shown in [8]. All Lie algebras considered here are finite dimensional over a field. Let