# Completeness of Low-Dimensional Leibniz Algebras

## Annual Meeting in Mathematics 2023

## Keywords:

Leibniz algebras, derivations, completeness, nilpotency, solvability## 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) \leq 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.

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## How to Cite

*Thai Journal of Mathematics*,

*22*(1), 165–178. Retrieved from https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1607