Analytic subgroup

The analytic subgroup is an important concept in mathematics (in Lie group theory). [Knapp, Anthony W. : Lie groups Beyond an Introduction, Second Edition.] [citebook|title=Theory of Lie Groups |author= Claude Chevalley|year= 1946|publisher=Princeton University Press|pages=99|id=ISBN 0691049904 |url=http://books.google.com/books?id=t-41JV63sRIC&pg=PA99&dq=%22Analytic+subgroup%22&ie=ISO-8859-1&output=html&sig=MVqzBeNtmDqt9XwsYKJCLl0kY24]

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 $i:H\; ightarrow\; G$ be the inclusion map. Then $di\_\{1\}\; :\; mathfrak\{h\}\; ightarrow\; mathfrak\{g\}$ 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.

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