Derivation Algebras of Three-Dimensional Leibniz Algebras: A Complete Description

Abstract

Derivation algebras are natural structural invariants of Leibniz algebras. In a number of earlier papers, derivations were described for several classes of one-generated, nilpotent, non-nilpotent, and low-dimensional Leibniz algebras. In the present paper we complete the description for three-dimensional non-Lie left Leibniz algebras over arbitrary fields. We use a refined organization of the three-dimensional classification into sixteen isomorphism types, including parameterized families. Previously known cases are incorporated by reference, with changes of basis recorded when necessary. For the remaining cases we determine the derivations by direct coefficient calculations. Special attention is paid to characteristic two, where several derivation algebras increase in dimension. We also record simple decompositions of the resulting Lie algebras into natural ideals and subalgebras. A final table gives the derivation algebra of every type in the classification.