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 groups, and, as shown by
George Mostow, for algebraic Lie algebras and simply connected
algebraic groups 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 ==