Lie Bracket for Group Elements of SU(3)

[nigelscott]

Homework Statement

Determine the Lie bracket for 2 elements of SU(3).

Homework Equations

[X,Y] = J_XY - J_YX where J are the Jacobean matrices

The Attempt at a Solution

I exponentiated λ₁ and λ₂ to get X and Y which are 3 x 3 matrices.. If the group elements are interpreted as vector fields then I ought to be able to apply the above equation to get Z (i..e. exp(iθλ₃). The problem is I don't know how to formulate the Jacobean matrices. Any help would be appreciated.

 

[Orodruin]

Determine the Lie bracket for 2 elements of SU(3).

SU(3) or su(3) (its Lie algebra)? The Lie bracket is an operation on the Lie algebra. Given general form of a rotation operator you should be able to find its composition with another and from there the commutator for small rotations.

 

[nigelscott]

I think I may be confusing myself. I think what you are saying is that although the commutator of 2 vector fields results in a third vector field on the manifold, that field at a given point is, by definition, assigned to a tangent space (as are the original fields). In this sense trying to figure out the commutator in terms of vectors fields on the manifold is not really the correct way to look at things or is a valid thing to do. All of the mathematics takes place in the tangent space. Am I getting close? I am new to this subject. Thanks.