On the decomposition of complete leibniz algebra

Abstract

Leibniz algebras, generalizations of Lie algebras, are characterized by their non-antisymmetric properties. In this study, we delve into the properties of decompositions within Leibniz algebras, drawing parallels with analogous results in Lie algebras. Our investigation extends to complete Leibniz algebras, focusing on the conditions governing their extensions. Similar to Lie algebras, we find that inner derivations play a pivotal role in characterizing complete Leibniz algebras. Specifically, it was revealed 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. Moreover, the quotient of the algebra of derivations of the Leibniz algebra by this ideal yields a complete Lie algebra. The results further demonstrated that any derivation of a semisimple Leibniz algebra can be expressed as a combination of three derivations. Additionally, the properties of the algebra of inner derivations were explored in comparison to the algebra of central derivations. We also delve into the study of generalizations of derivations of Leibniz algebras.

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