nLab
abelian Lie algebra

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Definition

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

[x,y]=0. [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 00-dimensional or 11-dimensional Lie algebra must be abelian. The 00-dimensional Lie algebra is the trivial Lie algebra. The 11-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)