nLab Lie ideal

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

References

See also:

• Groupprops, Ideal of a Lie ring