Ideal of a Lie algebra

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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

Definition with symbols

A subset B of a Lie algebra A is said to be an ideal if it is a vector subspace of A under addition, and [x,y] \in B for any x \in B and y \in A. 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 v and w is the differential at t = 0 of \exp(tv).w - w. Here \exp(tv) acts on w via the adjoint representation.

Now if w is an element in the Lie algebra of a closed normal subgroup N, then \exp(tv)\exp(t'w)\exp(tv)^{-1} is also in N for every t', and differentiating at t'=0 gives \exp{tv}.w to be in the Lie algebra of N. Thus, the difference \exp(tv).w - w is also in the Lie algebra, and hence so is the value [v,w]

Thus w being in the Lie algebra of N implies that [v,w] is. By the amticommutativity of the Lie bracket, we conclude that v being in the Lie algebra implies [v,w] is.