Completeness of low-dimensional leibniz algebras

Abstract

Leibniz algebras are generalizations of Lie algebras. By using the classification results of low-dimensional non-Lie nilpotent and non-nilpotent solvable Leibniz algebras obtained earlier, we define a basis of the derivation algebra Der(A) of each Leibniz algebra A and study their properties. It is known that for a Leibniz algebra A if the Lie algebra A/ Leib(A) is complete, then A is a complete Leibniz algebra. We show that the converse holds when A is a complete solvable Leibniz algebra with dim(A) ≤ 3. It is also known that for the derivation algebra of a complete Lie algebra is complete. However, our results show that this is not true for Leibniz algebras.