Lie group as manifold

  • #1
cianfa72
1,692
192
TL;DR Summary
About the Lie group parametrization as manifold
Hi,
consider the set of the following parametrized matrices

They are member of the group (indeed their determinant is 1). The group itself is homemorphic to a quadric in .

I believe the above parametrization is just a chart for the group as manifold. It makes sense only for and this condition yields an open subset in . On this open set the map is bijective.

That means the are matrices of that cannot be parametrized by the above map (i.e. matrices with the element in the first row/column equal 0).

My question is: the fact that the above parametrization does not cover entirely the group manifold does not rule out in principle that as manifold might be homeomorphic with . Thanks.
 
Last edited:
Physics news on Phys.org
  • #2
The question is a bit unclear. In any case is not homeomorphic to .
 
  • #3
martinbn said:
In any case is not homeomorphic to .
Ok, the point is: even if I found a parametrization (a chart) that doesn't cover entirely a manifold, there might be one that instead covers it entirely, right ?
 
  • #4
cianfa72 said:
Ok, the point is: even if I found a parametrization (a chart) that doesn't cover entirely a manifold, there might be one that instead covers it entirely, right ?
In general yes. Take an open proper subset of , it is a chart that doesn't cover the whole space, but there are charts that do.
 
  • Like
Likes cianfa72
  • #5
Ok, as in post #1 the Lie group is homeorphic with a quadric in with the subspace topology from -- in some sense this is "tautologically" true since the topology of is defined that way.

I believe we can cover (i.e. the quadric in ) with just 2 charts. One is the chart in post#1, the other one could be

Btw, as topological space should have the same topology as the subspace topology from . Indeed the latter is open in and of course is a subset of it. Then the open sets in are all and only the intersections of open sets in with the set .
 
Last edited:
  • #7
fresh_42 said:
- two charts.
This is confusing. The special linear group is not compact. It cannot be homeomorphic to a sphere. Also your notation for the unitary group is non standard. Do you mean or the complex points of the algebraic group?
 
  • #8
I just thought they should be "equal" for sharing the same Lie algebra, but you are right. And it's a mistake. The corresponding Lie algebras are only complex isomorphic.

It was so wrong, that I even missed the fact that the Lie algebra as the tangent space at is a local property.
 
  • Like
Likes martinbn
  • #9
Does make sense my post#5 ? Thank you.
 
  • #10
cianfa72 said:
Does make sense my post#5 ? Thank you.
Yes.
 
  • #11
cianfa72 said:
Ok, the point is: even if I found a parametrization (a chart) that doesn't cover entirely a manifold, there might be one that instead covers it entirely, right ?
That can happen if the manifold is globally, not just locally homeomorphic to .
 
  • #12
cianfa72 said:
Ok, the point is: even if I found a parametrization (a chart) that doesn't cover entirely a manifold, there might be one that instead covers it entirely, right ?
WWGD said:
That can happen if the manifold is globally, not just locally homeomorphic to .
It could happen even if it isn't homeomorphic to . For example take your manifold to be two disjoint open sets in then you have a global chart but it is not connected so it is not homeomorphic to the whole .
 
  • Like
Likes cianfa72
  • #13
martinbn said:
For example take your manifold to be two disjoint open sets in then you have a global chart but it is not connected so it is not homeomorphic to the whole .
Yes, the global chart map in this case is just the Identity map.
 

Similar threads

  • Topology and Analysis
Replies
16
Views
455
  • Topology and Analysis
2
Replies
61
Views
895
  • Topology and Analysis
Replies
8
Views
1K
  • Differential Geometry
Replies
4
Views
2K
  • Topology and Analysis
Replies
29
Views
5K
  • Topology and Analysis
Replies
1
Views
709
  • Topology and Analysis
2
Replies
54
Views
5K
  • Differential Geometry
Replies
17
Views
3K
  • Advanced Physics Homework Help
Replies
14
Views
1K
  • Linear and Abstract Algebra
Replies
4
Views
2K
Back
Top