∞-Lie theory

# Contents

## Definition

A Lie algebra $\mathfrak{g}$ is abelian if its bracket is identically 0, in that for all $x,y \in \mathfrak{g}$ we have

$[x,y] = 0 \,.$

## Examples

Every vector space has a (necessarily unique) abelian Lie algebra structure. As such, we may identify an abelian Lie algebra with its underlying vector space.

A $0$-dimensional or $1$-dimensional Lie algebra must be abelian. The $0$-dimensional Lie algebra is the trivial Lie algebra. The $1$-dimensional Lie algebra is a simple object in LieAlg, but it is traditionally not considered a simple Lie algebra.

## Lie integration

Under Lie integration abelian Lie algebras integrate to abelian Lie group?s.

Revised on September 6, 2010 09:38:44 by Urs Schreiber (134.100.32.213)