Lie subgroup

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.

References

External links

* [http://mathworld.wolfram.com/LieGroup.html Lie Group on Mathworld]

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