Levi-Mal'tsev decomposition
The presentation of a finite-dimensional Lie algebra over a field of characteristic zero as a direct sum (as vector spaces) of its radical
(the maximal solvable ideal in
) and a semi-simple Lie subalgebra
. 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
; moreover, the subalgebra
is unique up to an automorphism of the form
, where
is the inner derivation of the Lie algebra
determined by an element
of the nil radical (the largest nilpotent ideal) of
. If
is a connected and simply-connected real Lie group, then there are closed simply-connected analytic subgroups
and
of
, where
is the maximal connected closed solvable normal subgroup of
,
is a semi-simple subgroup of
,
, such that the mapping
,
,
, is an analytic isomorphism of the manifold
onto
; in this case the decomposition
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)) |
[4] | A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) |
[5] | M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) |
Comments
The existence of , called the Levi factor (if
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 holds for an algebraic group
. In this case
is the maximal unipotent normal subgroup and
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) |
[a2] | G. Hochschild, "The structure of Lie groups" , Holden-Day (1965) |
Levi-Mal'tsev decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Levi-Mal%27tsev_decomposition&oldid=13544