nLab
Lie ideal

Contents

Context

Lie theory

∞-Lie theory (higher geometry)

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

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

[V,𝔤]V [V,\mathfrak{g}] \subset V

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

References

See also

Created on September 25, 2017 at 08:43:28. See the history of this page for a list of all contributions to it.