In
Lie theory
In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact (mathematics), contact of spheres that have come to be called Lie theory. For instance, ...
and
representation theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
, 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 M ...
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 geometry. He ...
and proved by , states that any finite-dimensional
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
''g'' over a field of
characteristic zero 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
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product:
* an ''inner'' sem ...
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 algebra
In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...
s and is called the
Wedderburn principal theorem.
Extensions of the results
In representation theory, Levi decomposition of
parabolic subgroups 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 group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
s, 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 that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
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 algebra In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody ...
s 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