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 over a field of characteristic zero can be expressed as the semidirect product of a solvable ideal – namely its radical, the maximal solvable ideal of the algebra – and a semisimple subalgebra, often referred to as a Levi subalgebra.

The statement is sometimes phrased as: every finite‑dimensional Lie algebra g over a field of characteristic zero is the semidirect product

[ \mathfrak g = \mathfrak r \rtimes \mathfrak s, ]

where 𝔯 is a solvable ideal (the radical) and 𝔰 is a semisimple subalgebra (a Levi subalgebra).

When the semisimple part 𝔰 is regarded as a quotient of 𝔤, it is called the Levi factor of 𝔤. In practice this decomposition allows one to separate the “solvable” and “semisimple” behaviours of Lie algebras, thereby simplifying many classification problems that would otherwise require treating the whole structure at once.

The decomposition is not unique in the strict sense of set equality, but a deep theorem due to Malcev (1942) shows that any two Levi subalgebras of a given Lie algebra are conjugate via an inner automorphism of the form

[ \exp!\bigl(\operatorname{ad}(z)\bigr),\qquad z\in\mathfrak n, ]

where 𝔫 denotes the nilradical (the maximal nilpotent ideal) of 𝔤; this result is known as the Levi–Malcev theorem.

An analogous decomposition holds for associative algebras and is termed the Wedderburn principal theorem ([Associative algebra ] and [Wedderburn principal theorem ]).

Extensions of the results

In the broader setting of representation theory, the Levi decomposition of parabolic subgroups ([Borel_subgroup ]) of a reductive algebraic group plays a crucial role in the construction of induced (or parabolically induced) representations. The Langlands decomposition ([Langlands_decomposition ]), a refinement of the Levi decomposition for such subgroups, separates a parabolic subgroup into a product of a Levi component and a unipotent radical, thereby facilitating the induction process.

The theory extends naturally to simply connected Lie groups and, as demonstrated by George Mostow , also to algebraic Lie algebras and algebraic groups ([Algebraic_group ]) over fields of characteristic zero. In these contexts the Levi decomposition respects the topological and algebraic structures, allowing one to transfer results between the group and its Lie algebra.

However, the decomposition fails in many infinite‑dimensional settings. For instance, affine Lie algebras ([[Affine_Lie_algebra]]) possess a radical consisting solely of their centre, yet they cannot be expressed as a semidirect product of that centre with any complementary subalgebra. Likewise, over fields of positive characteristic, finite‑dimensional Lie algebras often lack a Levi decomposition altogether.

See also

• [Lie group decompositions ]