# Contents

## Definition

A Lie algebra $\mathfrak{g}$ is called reductive if the following equivalence 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.

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)

## References

Last revised on February 26, 2018 at 14:50:24. See the history of this page for a list of all contributions to it.