In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing[1] and Élie Cartan[2] and proved by Eugenio Elia Levi (1905), states that any finite-dimensional Lie algebra g over a field of characteristic zero is the semidirect product of a solvable ideal and a semisimple subalgebra.
Moreover, Malcev (1942) showed that any two Levi subalgebras are conjugate by an (inner) automorphism of the form where z is in the nilradical (Levi–Malcev theorem).
An analogous result is valid for associative algebras and is called the Wedderburn principal theorem.
In representation theory, Levi decomposition of parabolic subgroups of a reductive group is needed to construct a large family of the so-called parabolically induced representations.
The Levi decomposition also fails for finite-dimensional algebras over fields of positive characteristic.