Lie Subgroup
In
mathematics
, a
subgroup
H
of a
Lie group
G
is a
Lie subgroup
if it is also a
submanifold
of
G
. According to
Cartan's theorem
, this is equivalent to
H
being a
closed subset
in the
topological structure
of
G
. Then the
Lie algebra
h
of
H
is a
Lie subalgebra
of the Lie algebra
g
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.
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