Levi-Mal'tsev decomposition

The presentation of a finite-dimensional Lie algebra $L$ over a field of characteristic zero as a direct sum (as vector spaces) of its radical $R$ (the maximal solvable ideal in $L$) and a semi-simple Lie subalgebra $S\subset L$. It was obtained by E.E. Levi [1] and A.I. Mal'tsev [2]. The Levi–Mal'tsev theorem states that there always is such a decomposition $L=R+S$; moreover, the subalgebra $S$ is unique up to an automorphism of the form $\exp(\operatorname{ad}z)$, where $\operatorname{ad}z$ is the inner derivation of the Lie algebra $L$ determined by an element $z$ of the nil radical (the largest nilpotent ideal) of $L$. If $G$ is a connected and simply-connected real Lie group, then there are closed simply-connected analytic subgroups $R$ and $S$ of $G$, where $R$ is the maximal connected closed solvable normal subgroup of $G$, $S$ is a semi-simple subgroup of $G$, $R\cap S=\{e\}$, such that the mapping $(r,s)\to rs$, $r\in R$, $s\in S$, is an analytic isomorphism of the manifold $R\times S$ onto $G$; in this case the decomposition $G=RS=SR$ is also called a Levi–Mal'tsev decomposition.

References

[1]	E.E. Levi, Atti. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. , 40 (1906) pp. 3–17
[2]	A.I. Mal'tsev, "On the representation of an algebra as a direct sum of the radical and a semi-simple subalgebra" Dokl. Akad. Nauk SSSR , 36 : 2 (1942) pp. 42–45 (In Russian)
[3]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) MR0148716 MR0143793 Zbl 0121.27504 Zbl 0109.26201
[4]	A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) MR0412321 Zbl 0342.22001
[5]	M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018

Comments

The existence of $S$, called the Levi factor (if $S$ is a semi-simple subalgebra, respectively a semi-simple subgroup, also called a Levi subalgebra, respectively Levi subgroup), was established by Levi. The conjugacy of Levi factors was proved by Mal'tsev.

An analogue of the decomposition $G=RS$ holds for an algebraic group $G$. In this case $R$ is the maximal unipotent normal subgroup and $S$ is a maximal reductive subgroup (Mostow's theorem).

References

[a1]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002
[a2]	G. Hochschild, "The structure of Lie groups" , Holden-Day (1965) MR0207883 Zbl 0131.02702