User:Maximilian Janisch/latexlist/Algebraic Groups/Lie algebra, reductive

This page is a copy of the article Lie algebra, reductive in order to test automatic LaTeXification. This article is not my work.

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 $8$ is reductive is equivalent to any of the following properties:

1) the radical $r ( g )$ of $8$ coincides with the centre $3 ( \mathfrak { g } )$;

2) $\mathfrak { g } = \mathfrak { z } ( \mathfrak { g } ) \dot { + } \mathfrak { g } 0$, where $80$ is a semi-simple ideal of $8$;

3) $\mathfrak { g } = \sum _ { i = 1 } ^ { k } \mathfrak { g } _ { i }$, where the $93$ are prime ideals;

4) $8$ 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 $k = R$ 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 $h$ of a finite-dimensional Lie algebra $8$ over $k$ is said to be reductive in $8$ if the adjoint representation $: \mathfrak { h } \rightarrow \mathfrak { g } ( \mathfrak { g } )$ is completely reducible. In this case $h$ is a reductive Lie algebra. If $k$ is algebraically closed, then for a subalgebra $h$ of $8$ to be reductive it is necessary and sufficient that $r ( b )$ 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