Levi decomposition

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. One is its radical, a maximal solvable ideal, and the other is a semisimple subalgebra, called a Levi subalgebra. The Levi decomposition implies that any finite-dimensional Lie algebra is a semidirect product of a solvable Lie algebra and a semisimple Lie algebra.

When viewed as a factor-algebra of g, this semisimple Lie algebra is also called the Levi factor of g. To a certain extent, the decomposition can be used to reduce problems about finite-dimensional Lie algebras and Lie groups to separate problems about Lie algebras in these two special classes, solvable and semisimple.

Moreover, Malcev (1942) showed that any two Levi subalgebras are conjugate by an (inner) automorphism of the form

${\displaystyle \exp(\mathrm {ad} (z))\ }$

where z is in the nilradical (Levi–Malcev theorem).

An analogous result is valid for associative algebras and is called the Wedderburn principal theorem.