# nLab reductive Lie algebra

Contents

### Context

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## Definition

A Lie algebra $\mathfrak{g}$ is called reductive if the following equivalent conditions hold:

1. it is the direct sum $\mathfrak{g} \simeq \mathfrak{h} \oplus \mathfrak{a}$ of a semisimple Lie algebra $\mathfrak{h}$ and an abelian Lie algebra $\mathfrak{a}$;

2. its adjoint representation is completely reducible: every invariant subspace has an invariant complement.

Over a field of characteristic zero, the following conditions on $\mathfrak{g}$ are also equivalent to $\mathfrak{g}$ being reductive:

1. the radical of $\mathfrak{g}$ is equal to the centre of $\mathfrak{g}$ (in general, the radical is only contained inside the centre);

2. $\mathfrak{g}$ is a direct sum of its centre with a semisimple ideal;

3. $\mathfrak{g}$ is a direct sum of prime ideals.

More generally:

###### Definition

(Lie algebra reductive in ambient Lie algebra)

A sub-Lie algebra

$\mathfrak{h} \hookrightarrow \mathfrak{g}$

is called reductive if the adjoint Lie algebra representation of $\mathfrak{h}$ on $\mathfrak{g}$ is reducible.

(Koszul 50, recalled in e.g. Solleveld 02, def. 2.27)