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.

[edit] References

  1. ^ Knapp, Anthony W. : Lie groups Beyond an Introduction, Second Edition.
  2. ^ Claude Chevalley (1946). Theory of Lie Groups. Princeton University Press, 99. ISBN 0691049904. 
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