In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing[1] and Élie Cartan[2] and proved by Eugenio Elia Levi (1905), states that any finite-dimensional real[clarification needed]{Change real Lie algebra to a Lie algebra over a field of characteristic 0} 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.
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.
^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
and 27 Related for: Levi decomposition information
representation theory, the Levidecomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by Eugenio Elia Levi (1905), states that any finite-dimensional...
Jordan–Chevalley decomposition of an element in algebraic group as a product of semisimple and unipotent elements The Bruhat decomposition G = BWB of a semisimple...
{\mathfrak {a}},} while if the radical equals the center the Levidecomposition yields a decomposition g = s 0 ⊕ z ( g ) . {\displaystyle {\mathfrak {g}}={\mathfrak...
semidirect product of a solvable Lie group and a semisimple Lie group (the Levidecomposition). The classification of representations of solvable Lie groups is...
three papers in group theory: in the first one, Levi (1905) discovered what is now called Levidecomposition, which was conjectured by Wilhelm Killing and...
two large and generally complementary classes, as is shown by the Levidecomposition. The solvable Lie algebras are precisely those that can be obtained...
is zero. The significance of semisimplicity comes firstly from the Levidecomposition, which states that every finite dimensional Lie algebra is the semidirect...
{\displaystyle K({\mathfrak {g}},[{\mathfrak {g}},{\mathfrak {g}}])=0.} The Levidecomposition asserts that every finite-dimensional Lie algebra over a field of...
simply connected) and their compactness. A first key result is the Levidecomposition, which says that every simply connected Lie group is the semidirect...
algebra is reductive if and only if its radical equals its center. Levidecomposition Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2010), Algebras...
in g {\displaystyle {\mathfrak {g}}} ). This is in contrast to the Levidecomposition: the short exact sequence 0 → r a d ( g ) → g → g s s → 0 {\displaystyle...
Killing horizon Killing spinor Killing tensor Killing vector field Levidecomposition G2 (mathematics) Root system O'Conner, J. J.; Robertson, E. F. (February...
R\to 1.} If k has characteristic zero, then one has the more precise Levidecomposition: every connected linear algebraic group G over k is a semidirect product...
Lie algebras include Whitehead's lemmas, Weyl's theorem, and the Levidecomposition theorem. Let g {\displaystyle {\mathfrak {g}}} be a Lie algebra over...
field or rotation field. This decomposition does not exist for all vector fields and is not unique. The Helmholtz decomposition in three dimensions was first...
shortage of technical methods, often inductive in nature and based on Levidecompositions amongst other matters, but the field was – and is – very demanding...
Paul Levi (11 March 1883 – 9 February 1930) was a German communist and social democratic political leader. He was the head of the Communist Party of Germany...
decomposition or rank-R decomposition is the decomposition of a tensor as a sum of R rank-1 tensors, where R is minimal. Computing this decomposition...
defined from its decomposition into the product of transpositions as sgn(σ) = (−1)m where m is the number of transpositions in the decomposition. Although such...
spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, it was used by Albert Einstein to develop his general theory of relativity...
group that is invariant under inner automorphisms. Iwasawa decomposition Iwasawa decomposition Jacobi identity 1. Carl Gustav Jacob Jacobi (1804 – 1851)...
any n × n rotation matrix. In fact, we can view the sequential angle decomposition, discussed previously, as reversing this process. The composition of...
Yateley Heath Woods in Yateley, Hampshire, on 18 September. On 23 June 2011, Levi Bellfield, already subject to three life sentences with a whole life tariff...
possessing some kind of symmetry. Important examples include: holonomy of the Levi-Civita connection in Riemannian geometry (called Riemannian holonomy), holonomy...
{\color {green}{k}} )\\\end{aligned}}} This can be interpreted as the decomposition of a × b into the sum of nine simpler cross products involving vectors...
distinct cycles involve disjoint sets of elements, this is referred to as "decomposition into disjoint cycles". To write down the permutation σ {\displaystyle...
fields are so ubiquitous that they are often simply called "tensors". Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 – continuing...