Lie group multiplication and Lie algebra commutation

1,340
30
I've heard it said that the commutation relations of the generators of a Lie algebra determine the multiplication laws of the Lie group elements.

I would like to prove this statement for ##SO(3)##.

I know that the commutation relations are ##[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}##.

Can you suggest a possible next step for showing how this can be used to determine the multiplication law for ##SO(3)##?
 
578
18
I'm not used to Lie but I think the steps are : You could find a set of matrices that satisfy these commutation relation, $$J_k $$

Then it builds a basis for the Lie algebra.

By exponentiating we get elements of the group SO (3)

Thus we need to compute $$exp (aJ_x+bJ_z)$$ for exemple.
 
1,340
30
Does the multiplication law refer to the fact that if we multiply two elements ##e^{i\theta_{1}J_{1}}## and ##e^{i\theta_{2}J_{2}}## from the Lie group, we get an element which is also in the Lie group?

Or does the group multiplication law refer to the fact that ##e^{i\theta_{1}J_{1}}e^{i\theta_{2}J_{2}}=e^{i(\theta_{1}J_{1}+\theta_{2}J_{2})}##?
 
578
18
Does the multiplication law refer to the fact that if we multiply two elements ##e^{i\theta_{1}J_{1}}## and ##e^{i\theta_{2}J_{2}}## from the Lie group, we get an element which is also in the Lie group?

Or does the group multiplication law refer to the fact that ##e^{i\theta_{1}J_{1}}e^{i\theta_{2}J_{2}}=e^{i(\theta_{1}J_{1}+\theta_{2}J_{2})}##?
The last is in general not true see Baker Campbell Hausdorff formula
 
1,340
30
Alright then, how would you define the multiplication law for ##SO(3)##?
 
578
18
The operation could be matrix multiplication but the exponential comes from writing a rotation out of an infinitesimal one.

Infinitesimal rotations are commutative but rotations are not.
 

Related Threads for: Lie group multiplication and Lie algebra commutation

Replies
3
Views
782
Replies
7
Views
764
  • Posted
Replies
12
Views
5K
  • Posted
Replies
5
Views
2K
Replies
4
Views
2K
  • Posted
Replies
8
Views
1K
  • Posted
Replies
1
Views
2K
  • Posted
Replies
11
Views
1K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top