Lie Algebra Structure of 3-Sphere: Left Invariant Vector Fields

[wofsy]

I know I should be able to look this up but am having trouble this morning.

I would like to know the Lie Algebra structure of the three sphere. In particular I'd like to express it in terms of the left invariant vector fields that are either tangent to the fibers of the Hopf fibration or perpendicular to it. By this I mean express the algebra in terms if an ortho-normal left invariant basis one of whose elements is tangent to the fiber circles of the Hopf fibration.

 

[morphism]

The Lie group structure on S^3 is that of SU(2), so the Lie algebra of S^3 is su(2).

I don't know if this will help you get the explicit construction you're after, but there is a 2:1 covering SU(2) -> SO(3) from which one can obtain the Hopf fibration.

 

[wofsy]

thanks that's very helpful

I think it won't be hard to break this out in terms of the Hopf fibration. In fact I think this is what they do when describing electron spin in quantum mechanics.

An aside question that I have is - can such a basis be viewed as determining a Riemannian connection on S3 when viewed as the unit circle bundle of a 2-plane bundle over the 2 sphere? If so, then the Lie algebra structure should tell me the Gauss curvature.

Thanks again.

 

[zhentil]

Yes, I believe this would be a special case of the construction of the Berger metrics on S^3.