Nilradical Of A Lie Algebra
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In
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
, the nilradical of a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
is a nilpotent
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
, which is as large as possible. The nilradical \mathfrak(\mathfrak g) of a finite-dimensional Lie algebra \mathfrak is its maximal
nilpotent ideal In mathematics, more specifically ring theory, an ideal ''I'' of a ring ''R'' is said to be a nilpotent ideal if there exists a natural number ''k'' such that ''I'k'' = 0. By ''I'k'', it is meant the additive subgroup generated by the set of ...
, which exists because the sum of any two nilpotent ideals is nilpotent. It is an ideal in the
radical Radical may refer to: Politics and ideology Politics *Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
\mathfrak(\mathfrak) of the Lie algebra \mathfrak. The quotient of a Lie algebra by its nilradical is a reductive Lie algebra \mathfrak^. However, the corresponding
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
: 0 \to \mathfrak(\mathfrak g)\to \mathfrak g\to \mathfrak^\to 0 does not split in general (i.e., there isn't always a ''subalgebra'' complementary to \mathfrak(\mathfrak g) in \mathfrak). This is in contrast to the
Levi decomposition In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by , states that any finite-dimensional real Lie algebra ''g'' is the semidirect product of a solvable ideal and a semi ...
: the short exact sequence : 0 \to \mathfrak(\mathfrak g)\to \mathfrak g\to \mathfrak^\to 0 does split (essentially because the quotient \mathfrak^ is semisimple).


See also

*
Levi decomposition In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by , states that any finite-dimensional real Lie algebra ''g'' is the semidirect product of a solvable ideal and a semi ...
*
Nilradical of a ring In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements: :\mathfrak_R = \lbrace f \in R \mid f^m=0 \text m\in\mathbb_\rbrace. In the non-commutative ring case the same definition does not always work. Th ...
, a notion in ring theory.


References

* * {{citation, title=Lie Groups and Lie Algebras III: Structure of Lie Groups and Lie Algebras, first1=Arkadi L., last1=Onishchik, author1-link=Arkadi L. Onishchik, first2=Ėrnest Borisovich, last2= Vinberg, author2-link=Ernest Borisovich Vinberg , publisher=Springer, year= 1994, isbn=978-3-540-54683-2. Lie algebras