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Lie subgroup

A Lie subgroup H {\displaystyle H} of a Lie group G {\displaystyle G} is a Lie group that is a subset of G {\displaystyle G} and such that the inclusion map from H {\displaystyle H} to G {\displaystyle G} is an injective immersion and group homomorphism. According to Cartan's theorem, a closed subgroup of G {\displaystyle G} admits a unique smooth structure which makes it an embedded Lie subgroup of G {\displaystyle 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 {\displaystyle G} to be a torus of dimension 2 or greater, and let H {\displaystyle 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: R → G {\displaystyle \varphi :\mathbb {R} \to G} with i m (φ) = H {\displaystyle \mathrm {im} (\varphi)=H}. The closure of H {\displaystyle H} will be a sub-torus in G {\displaystyle G}.

The exponential map gives a one-to-one correspondence between the connected Lie subgroups of a connected Lie group G {\displaystyle G} and the subalgebras of the Lie algebra of G {\displaystyle G}. Typically, the subgroup corresponding to a subalgebra is not a closed subgroup. There is no criterion solely based on the structure of G {\displaystyle G} which determines which subalgebras correspond to closed subgroups.