Ideal of a Lie algebra
This article defines a property for a subalgebra of a Lie algebra
ANALOGY: This is an analogue in Lie algebra of a property encountered in group. Specifically, it is a Lie subalgebra property analogous to the subgroup property: normal subgroup
View other analogues of normal subgroup | View other analogues in Lie algebras of subgroup properties (OR, View as a tabulated list)
Further information: Ideal of a Lie ring
Definition with symbols
A subset of a Lie algebra is said to be an ideal if it is a vector subspace of under addition, and for any and . Note that any ideal is, in particular, also a Lie subalgebra.
Relation with the Lie group
The Lie algebra of any closed normal subgroup of a Lie group, is an ideal. This follows from the following:
The commutator of and is the differential at of . Here acts on via the adjoint representation.
Now if is an element in the Lie algebra of a closed normal subgroup , then is also in for every , and differentiating at gives to be in the Lie algebra of . Thus, the difference is also in the Lie algebra, and hence so is the value
Thus being in the Lie algebra of implies that is. By the amticommutativity of the Lie bracket, we conclude that being in the Lie algebra implies is.