Isomorphism between groups and their Lie Algebra

[raopeng]

I must apologize if this question sounds dump but if an isomorphism is established between two groups, is it true that their lie algebra is an isomorphism too? My idea is that since the tangent space is sent to tangent space also in the matrix space, both groups' lie algebra will be isomorphism also. For example we have: SU(2) × SU(2) ≈ SO(4) and su(2) × su(2) ≈ so(4).

 

[GoodSpirit]

Hi raopeng,

Could you explain a little bit more?

Best regards

GoodSpirit

 

[raopeng]

For example we have an isomorphism between two groups, and we know that Lie Algebra of a group is the commutator of matrix of its tangent space at the identity. So if there exists an isomorphism, wouldn't that entail that, since the tangent space is sent to tangent space under the mapping, their lie algebra is isomorphic too? I have this idea when trying to establish an isomorphism between su(2) x su(2) ≈ so(4) and there is an isomorphism between SU(2) x SU(2) ≈ SO(4)

 

[wisvuze]

A lie group isomorphism f between lie groups G and H will have full rank, so that the corresponding map between lie algebras df is an isomorphism. However, the converse is false, two lie groups can have the same lie algebra but be non isomorphic lie groups.
Groups that are locally diffeomorphic at the identity will have isomorphic lie algebras

 

[lavinia]

I must apologize if this question sounds dump but if an isomorphism is established between two groups, is it true that their lie algebra is an isomorphism too? My idea is that since the tangent space is sent to tangent space also in the matrix space, both groups' lie algebra will be isomorphism also. For example we have: SU(2) × SU(2) ≈ SO(4) and su(2) × su(2) ≈ so(4).

you need to show that df[X,Y] = [df(X),dfY)] which implies that the linear isomorphism of left invariant vector fields is a Lie algebra homomorphism.