In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by Eugenio Elia Levi (1905), states that any finite-dimensional real Lie algebra g is the semidirect product of a solvable ideal and a semisimple 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.
|Conjectured by||Wilhelm Killing|
|First proof by||Eugenio Elia Levi|
|First proof in||1905|
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.
where z is in the nilradical (Levi–Malcev 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.
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.
- Killing, W. (1888). "Die Zusammensetzung der stetigen endlichen Transformationsgruppen". Mathematische Annalen. 31 (2): 252–290. doi:10.1007/BF01211904.
- Cartan, Élie (1894), Sur la structure des groupes de transformations finis et continus, Thesis, Nony
- Jacobson, Nathan (1979). Lie algebras. New York: Dover. ISBN 0486638324. OCLC 6499793.
- Levi, Eugenio Elia (1905), "Sulla struttura dei gruppi finiti e continui", Atti della Reale Accademia delle Scienze di Torino. (in Italian), XL: 551–565, JFM 36.0217.02, archived from the original on March 5, 2009 Reprinted in: Opere Vol. 1, Edizione Cremonese, Rome (1959), p. 101.
- Maltsev, Anatoly I. (1942), "On the representation of an algebra as a direct sum of the radical and a semi-simple subalgebra", C. R. (Doklady) Acad. Sci. URSS (N.S.), 36: 42–45, MR 0007397, Zbl 0060.08004.