In
Lie theory
In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is ...
and
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, the Levi decomposition, conjectured by
Wilhelm Killing
Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry.
Life
Killing studied at the University of Mü ...
and
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
and proved by , states that any finite-dimensional real
Lie algebra ''g'' is the semidirect product of a
solvable ideal and a
semisimple
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
subalgebra.
One is its
radical, a maximal solvable ideal, and the other is a semisimple subalgebra, called a Levi subalgebra. The Levi decomposition implies that any finite-dimensional Lie algebra is a
semidirect product of a solvable Lie algebra and a semisimple Lie algebra.
When viewed as a factor-algebra of ''g'', this semisimple Lie algebra is also called the Levi factor of ''g''. To a certain extent, the decomposition can be used to reduce problems about finite-dimensional Lie algebras and Lie groups to separate problems about Lie algebras in these two special classes, solvable and semisimple.
Moreover,
Malcev (1942) showed that any two Levi subalgebras are
conjugate by an (inner) automorphism of the form
:
where ''z'' is in the
nilradical (Levi–Malcev theorem).
An analogous result is valid for
associative algebras and is called the
Wedderburn principal theorem.
Extensions of the results
In representation theory, Levi decomposition of
parabolic subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgro ...
s of a reductive group is needed to construct a large family of the so-called
parabolically induced representations. The
Langlands decomposition is a slight refinement of the Levi decomposition for parabolic subgroups used in this context.
Analogous statements hold for simply connected
Lie groups, and, as shown by
George Mostow, for algebraic Lie algebras and simply connected
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Ma ...
s over a field of
characteristic zero.
There is no analogue of the Levi decomposition for most infinite-dimensional Lie algebras; for example
affine Lie algebras have a radical consisting of their center, but cannot be written as a semidirect product of the center and another Lie algebra. The Levi decomposition also fails for finite-dimensional algebras over fields of positive characteristic.
See also
*
Lie group decompositions
References
Bibliography
*
* Reprinted in: Opere Vol. 1, Edizione Cremonese, Rome (1959), p. 101.
*.
External links
*{{springer, id=Levi%E2%80%93Mal%27tsev_decomposition&oldid=32912, title=Levi-Mal'tsev decomposition, author=A.I. Shtern
Lie algebras