Levi decomposition - Imported Knowlpedia Content

Let $\mathfrak g$ be a finite-dimensional Lie algebra over a field of characteristic $0$ .

Theorem (Levi decomposition). There exists a largest solvable ideal $\mathfrak r\subseteq \mathfrak g$ , called the radical (a notion built from solvability and ideals as in ideal), and a semisimple subalgebra $\mathfrak s\subseteq \mathfrak g$ such that

\mathfrak g \cong \mathfrak s \ltimes \mathfrak r

as Lie algebras. Here $\mathfrak s$ is called a Levi factor and $\mathfrak r$ is the solvable radical.

Remarks

Moreover, any two Levi factors are conjugate by an inner automorphism of $\mathfrak g$ (more precisely, by an automorphism arising from the exponential of an inner derivation coming from $\mathfrak r$ ), so the semisimple part is essentially unique.

Context. This theorem isolates the “semisimple core” of a Lie algebra and reduces many problems to understanding semisimple algebras (see semisimple Lie algebras) plus solvable/nilpotent structure (compare derived series and nilpotent Lie algebras). It is also a key input in analyzing Lie algebras arising from compact Lie groups.