Lie subgroup

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In mathematics, a Lie subgroup H of a Lie group G is a Lie group that is a subset of G and such that the inclusion map from H to G is an injective immersion and group homomorphism. According to Cartan's theorem, a closed subgroup of G admits a unique smooth structure which makes it an embedded Lie subgroup of G -- i.e. a Lie subgroup such that the inclusion map is a smooth embedding.

Examples of non-closed subgroups are plentiful; for example take G to be a torus of dimension ≥ 2, and let H be a one-parameter subgroup of irrational slope, i.e. one that winds around in G. Then there is a Lie group homomorphism φ : RG with H as its image. The closure of H will be a sub-torus in G.

In terms of the exponential map of G, in general, only some of the Lie subalgebras of the Lie algebra g of G correspond to closed Lie subgroups H of G. There is no criterion solely based on the structure of g which determines which those are.


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