A characterization of the semisimplity of Lie-type algebras through the existence of certain linear bases

Abstract

The class of Lie-type algebras contains the ones of associative algebras, Lie algebras and Leibniz algebras among other classes of algebras. We show that a Lie-type algebra , of arbitrary dimension and over an arbitrary base field, is semisimple if and only if it has zero annihilator and admits a weak-division linear basis. As a corollary, the simplicity of is also characterized.