On Inner Derivations of Leibniz Algebras

Leibniz algebras are generalizations of Lie algebras. Similar to Lie algebras, inner derivations play a crucial role in characterizing complete Leibniz algebras. In this work, we demonstrate that the algebra of inner derivations of a Leibniz algebra can be decomposed into the sum of the algebra of left multiplications and a certain ideal. Furthermore, we show that the quotient of the algebra of derivations of the Leibniz algebra by this ideal yields a complete Lie algebra. Our results independently establish that any derivation of a semisimple Leibniz algebra can be expressed as a combination of three derivations. Additionally, we compare the properties of the algebra of inner derivations of Leibniz algebras with the algebra of central derivations.