Levi DecompositionEdit

Levi decomposition is a foundational result in the structure theory of Lie algebras. Over a field of characteristic zero, every finite-dimensional Lie algebra g can be expressed as a semidirect product g = s ⋉ r, where r is the radical (the largest solvable ideal) and s is a Levi subalgebra that is semisimple. Named after Eugenio Elia Levi, this theorem reveals how a Lie algebra’s structure splits into a robust semisimple core and a solvable extension, with the two pieces interacting in a controlled way.

In practice, the Levi decomposition provides a clean lens for understanding representations, automorphisms, and the behavior of a Lie algebra under various constructions. It is a bridge between the theory of semisimple semisimple Lie algebras, which are well-understood and rigid, and the broader world of solvable and solvable-by-semisimple structures. The theorem is also a backbone for the parallel theory of algebraic groups, where a similar decomposition exists for connected linear groups, tying Lie algebra structure to group-theoretic properties.

Historical background

The decomposition is classically attributed to Levi, who established the result in the early 20th century. In the language of modern algebra, the Levi-Malcev theorem solidifies the parallel between the Lie algebra setting and its group-theoretic counterpart. Subsequent work clarified the boundaries of the theorem, especially in relation to the characteristic of the ground field and to various generalizations, including algebraic groups in characteristic zero and beyond. See Levi-Malcev theorem for a broader view of these ideas and their consequences.

Statement of the Levi decomposition

Let g be a finite-dimensional Lie algebra over a field F of characteristic zero. Then there exists a subalgebra s ⊆ g such that:

s is semisimple lie algebra, i.e., has no nontrivial solvable ideals,
r = rad(g) is the radical (the largest solvable ideal) of g, and
g = s ⋉ r as a Lie algebra, i.e., the bracket satisfies [s, s] ⊆ s, [s, r] ⊆ r, and [r, r] ⊆ r.

Moreover, any two Levi subalgebras s1 and s2 of g are conjugate by an inner automorphism of g. In particular, while the decomposition is not canonical, the semisimple part is unique up to conjugacy inside g, and the radical is uniquely determined by g.

The components in the decomposition have precise meaning:

the radical r captures all solvable behavior of g; it is an ideal that encodes the “nonsemisimple” part,
the Levi subalgebra s provides a semisimple backbone, reflecting the rigid, well-understood portion of the algebra,
the structure g ≅ s ⋉ r expresses g as a semidirect product: s acts on r by the adjoint action, and r remains a solvable ideal.

For a broader perspective, the theorem holds in characteristic zero, and its proof uses core results like Engel’s theorem and Malcev’s ideas about solvable and semisimple components. See Engel's theorem for related tools, and Levi-Malcev theorem for the historical articulation and generalizations.

Structure and consequences

The radical r is the maximal solvable ideal of g. This means any solvable substructure of g sits inside r, and r itself is closed under the Lie bracket and under the adjoint action of g.
The Levi factor s is semisimple, meaning it decomposes into a direct sum of simple Lie algebras, and it carries a rigid representation theory that is central to understanding g’s representations.
Conjugacy of Levi subalgebras: while different choices of s may appear, any two such Levi subalgebras are related by an inner automorphism of g. This gives a robust notion of “the semisimple part” of g, even though there is no unique canonical Levi subalgebra.
Representation theory: the decomposition guides the study of g-modules by separating the action of a semisimple part from the solvable radical. In particular, the action of s on r plays a pivotal role in determining the structure of representations.
Connections to reductive and solvable structures: reductive Lie algebras (those that are a direct sum of a semisimple part and an abelian center) sit naturally inside this framework, with the radical often reflecting central or near-central features.

Examples

gl_n has a classical Levi decomposition gl_n ≅ sl_n ⋉ center(gl_n). The radical is the center (scalar matrices), which is a one-dimensional solvable ideal, and the Levi subalgebra is sl_n, a semisimple Lie algebra. This concrete case shows the radical and Levi factor living in a familiar matrix setting.
- In this example, the center acts trivially on the radical, so the semidirect product is effectively a direct sum as a vector space, while the Lie bracket encodes the standard adjoint action of sl_n on the center.
- See gl_n and sl_n for the standard templates of these algebras.
More generally, if g = sl_k ⋉ V where V is a finite-dimensional representation of sl_k, then r = V is the radical and s = sl_k is a Levi subalgebra. This illustrates how a semisimple core can be extended by a solvable module to produce a wide class of nontrivial Lie algebras.
A semisimple Lie algebra, such as a direct sum of simple pieces, has radical r = 0, so the Levi decomposition reduces to g = s with s semisimple. This case highlights the extremes of the spectrum covered by the theorem.

Variants and generalizations

Algebraic groups: over algebraically closed fields of characteristic zero, connected linear algebraic groups admit a Levi decomposition into a semidirect product of a reductive subgroup and the unipotent radical. This mirrors the Lie algebra case on the level of the tangent space and group structure.
Positive characteristic: in fields of positive characteristic, Levi’s theorem is not guaranteed in full generality. Additional hypotheses are often required, and counterexamples exist. The study of Levi-type results in this setting leads to refined structure theories and introduces extra phenomena not present in characteristic zero.
Super Lie algebras and other generalizations: the Levi decomposition has been explored in broader contexts, including graded or super extensions, where analogous structure results provide insight into more elaborate symmetry theories.