1-parameter subgroup

For a Lie group , a 1-parameter subgroup is a Lie group morphism

Notes

• By definition, is a smooth curve on .
• The set of all 1-parameter subgroups is not a subgroup itself, however, the image of is a subgroup.
• does not need to be injective.

Correspondence between integral curves

Note that simply from the definition, 1-parameter subgroups are not integral curves. However, we can show that there is a strong correspondence between them 1-parameter subgroups and integral curves of left-invariant vector fields.

Proposition 1

The unique integral curve of with initial condition is a 1-parameter subgroup.

Proof

Let be the flow generated by . Then by we know that so looking at we see that it is a 1-parameter subgroup

Proposition 2

If is a 1-parameter subgroup, then it is the integral curve of a left-invariant vector field.

Proof

We know that is a flow (we need the minus sign to ensure that it is a left action). Note that

So the flow commutes with left multiplication so it is the flow of a left invariant vector field.
Take the integral curve of to finish the proof.