- Levi decomposition
In
Lie theory andrepresentation theory , the Levi decomposition, discovered byEugenio Elia Levi (1906), states that any finite dimensional realLie algebra "g" is (as a vector space) the direct sum of two significant structural parts; namely, solvable subalgebra and semisimple subalgebra.Formal statement
One is its radical, a maximal solvable ideal, and the other is a semisimple subalgebra, called a Levi subalgebra. 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". Moreover,Malcev (1942) showed that any two Levi subalgebras are conjugate by an (inner) automorphism of the form:exp(Ad("z"))
where "z" is in the
nilradical (Levi–Malcev theorem).Application
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.
Extensions of the results
In representation theory, Levi decomposition of
parabolic subgroup s of a reductive group is needed to construct a large family of the so-called parabolically induced representations. TheLanglands decomposition is a slight refinement of the Levi decomposition for parabolic subgroups used in this context.Analogous statements hold for simply connected
Lie group s, and, as shown byGeorge Mostow , for algebraic Lie algebras and simply connectedalgebraic group s over a field of characteristic zero. There is no analogue of the Levi decomposition for most infinite-dimensional Lie algebras; for exampleaffine Lie algebra s have a radical consisting of their center, but cannot be written as a semidirect product of the center and another Lie algebra.See also
*
Lie group decompositions Notes
References
*Jacobson, "Lie algebras"
*E.E. Levi, Atti. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 40 (1906) pp. 3–17
*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. 42–45External links
*springer|id=l/l058250|title=Levi-Mal'tsev decomposition|author=A.I. Shtern
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