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