Lie subgroup
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In mathematics, a subgroup H of a Lie group G is a Lie subgroup if the inclusion map from H to G is smooth. In particular, this implies that the inclusion map from H to G is an immersion. According to Cartan's theorem, a closed subgroup of G is always a Lie subgroup of G.
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 : R → G 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 Lie subgroups H of G. There is no criterion solely based on the structure of g which determines which those are.
[edit] References
- Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, 34, Providence, R.I.: American Mathematical Society, MR1834454, ISBN 978-0-8218-2848-9
