Docs, Lie group actions are non-commutative (i.e. non-symmetric)

[dehann]

trafficstars

Hi, I just want to confirm one line in the docs for exp_lie. I have Lie group actions in general as non-commutative?
https://github.com/JuliaManifolds/Manifolds.jl/blob/8274392cc6a1c406f7fe83012e2feefb85e58738/src/groups/group.jl#L846

My concern is that the composition operation here is not symmetric in the general case, or I am missing something?

I have in general that the sum of Lie algebra elements (tangent vectors about group identity in this case**?) are commutative up to the Lie bracket. As in, Baker-Cambell-Hausdorf (wiki and also this) formula:

exp(X)exp(Y) = exp(Z)

where

Z = X+Y + 0.5[X,Y] + ...

for non-commutative, the Lie bracket [X,Y] is non-zero.

---

** separate question on log for Group (Lie Algebra diagonals), see

• #521

[kellertuer]

Yes, I think that is a typo in general, the last part should be omitted.

[mateuszbaran]

See for example 3.5 of this: https://www.sciencedirect.com/topics/mathematics/one-parameter-subgroup , one-parameter subgroups of Lie groups are commutative.

[mateuszbaran]

@kellertuer should we make the docs more clear that this is actually correct?

[kellertuer]

I think that would be a good idea.

[dehann]

one-parameter subgroups of Lie groups are commutative.

got it thanks. Perhaps just a comment ("for commutative groups")?

[mateuszbaran]

What do you mean by "for commutative groups" here? Even if a Lie group isn't commutative, its one-parameter subgroups are commutative.

[dehann]

Ah, so this is the key point:

Even if a Lie group isn't commutative, its one-parameter subgroups are commutative

The docs do state "one parameter subgroup" on scalars s and t. I did not know that was sufficient for commutative.
https://github.com/JuliaManifolds/Manifolds.jl/blob/8274392cc6a1c406f7fe83012e2feefb85e58738/src/groups/group.jl#L840

So this issue then is more about my knowledge gaps than precision. My suggestion then would be perhaps just add: "Note, even if a Lie group isn't commutative, its one-parameter subgroups are commutative.".