# nLab Lie ideal

Contents

### Context

#### 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

Given a Lie algebra with underlying vector space $\mathfrak{g}$ and Lie bracket $[-,-]\colon \mathfrak{g} \otimes \mathfrak{g} \to \mathfrak{g}$, then an ideal is a sub-vector space $V \subset \mathfrak{g}$ such that

$[V,\mathfrak{g}] \subset V$

hence such that for all $v \in V \subset \mathfrak{g}$ and $x \in \mathfrak{g}$ one has $[v,x] \in V \subset \mathfrak{g}$.

This is sometimes referred to as a Lie ideal.

## Properties

A Lie ideal is always a sub-Lie algebra