Homomorphism between Lie groups

  • #1
694
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Hello I've been reading some Group theory texts and would like to clarify something.
Let's say we have two Lie groups A and B, with corresponding Lie algebras a and b.
Does the fact that a and b share the same Lie Bracket structure, as in if we can find a map
M:a->b which obeys [M(q),M(p)]=M([q,p]), mean that the two corresponding Lie groups are homomorphic to each other? (Due to the BCH formula)
 

Answers and Replies

  • #2
137
0
It's mathematics not physics !
 
  • #3
Matterwave
Science Advisor
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Two Lie groups which have the same algebra have the same structure around the identity element. But many groups with the same lie algebras will have different global properties (different structures away from the identity element). For example SU(2) and SO(3) have the same Lie algebra (the cross product algebra) but SU(2) is simply connected whereas SO(3) is not. In addition, some groups have disconnected components away from the identity which the Lie algebra is not sensitive to at all. For example SO(3) and O(3) would have identical Lie algebras because they are exactly the same at the identity; however, O(3) has a disconnected component (those orthogonal matrices with determinant -1) which the Lie algebra can't "see". So there is not necessarily a homeomorphism between two Lie groups which have the same algebras.
 

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