Analytic subgroup

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The analytic subgroup is an important concept in mathematics (in Lie group theory).^[1]^[2]

An analytic subgroup H of a Lie group G is a subgroup which is a connected Lie group, and for which the inclusion mapping is smooth and everywhere regular.

Note that the analytic subgroup H is itself an analytic group.

Let $\mathfrak{h}$ and $\mathfrak{g}$ be the Lie algebras of H and G, and let $i:H \rightarrow G$ be the inclusion map. Then $di_{1} : \mathfrak{h} \rightarrow \mathfrak{g}$ is an injective Lie algebra homomorphism: any analytic subgroup gives a Lie subalgebra in a natural way.

This can be seen as a map from analytic subgroups of G to Lie subalgebras of $\mathfrak{g}$ . A very important theorem in Lie theory is that this map is, in fact, bijective.