Lie algebra, reductive

A finite-dimensional Lie algebra over a field $k$ of characteristic 0 whose adjoint representation is completely reducible (cf. Adjoint representation of a Lie group; Representation of a Lie algebra). The property that a Lie algebra $\mathfrak g$ is reductive is equivalent to any of the following properties:

1) the radical $\mathfrak r ( \mathfrak g )$ of $\mathfrak g$ coincides with the centre $\mathfrak z ( \mathfrak g )$ ;

2) $\mathfrak g = \mathfrak z ( \mathfrak g ) \dot{+} \mathfrak g _{0}$ , where $\mathfrak g _{0}$ is a semi-simple ideal of $\mathfrak g$ ;

3) $\mathfrak g = \sum _{i=1} ^{k} \mathfrak g _{i}$ , where the $\mathfrak g _{i}$ are prime ideals;

4) $\mathfrak g$ admits a faithful completely-reducible finite-dimensional linear representation.

The property that a Lie algebra is reductive is preserved by both extension and restriction of the ground field $k$ .

An important class of reductive Lie algebras over $k = \mathbf R$ are the compact Lie algebras (see Lie group, compact). A Lie group with a reductive Lie algebra is often called a reductive Lie group. A Lie algebra over $k$ is reductive if and only if it is isomorphic to the Lie algebra of a reductive algebraic group over $k$ .

A generalization of the concept of a reductive Lie algebra is the following. A subalgebra $\mathfrak h$ of a finite-dimensional Lie algebra $\mathfrak g$ over $k$ is said to be reductive in $\mathfrak g$ if the adjoint representation $\mathop{\rm ad}\nolimits : \ \mathfrak h \rightarrow \mathfrak g \mathfrak l ( \mathfrak g )$ is completely reducible. In this case $\mathfrak h$ is a reductive Lie algebra. If $k$ is algebraically closed, then for a subalgebra $\mathfrak h$ of $\mathfrak g$ to be reductive it is necessary and sufficient that $\mathop{\rm ad}\nolimits \ \mathfrak r ( \mathfrak h )$ consists of semi-simple linear transformations.

References

 [1] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803 [2] J.-P. Serre, "Algèbres de Lie semi-simples complexes" , Benjamin (1966) MR0215886 Zbl 0144.02105 [3] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002
How to Cite This Entry:
Lie algebra, reductive. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_reductive&oldid=44265
This article was adapted from an original article by r group','../u/u095350.htm','Unitary transformation','../u/u095590.htm','Vector bundle, analytic','../v/v096400.htm')" style="background-color:yellow;">A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article