Why Do My Calculations of Structure Constants in SO(3) Differ from Textbooks?

[marton]

The following matrices are written in Matlab codes form.

The standard basis for so(3) is: L1 = [0 0 0; 0 0 -1; 0 1 0], L2 = [0 0 1; 0 0 0; -1 0 0], L3 = [0 -1 0; 1 0 0; 0 0 0]. Since [L1, L2] = L3, the structure constants of this Lie algebra are C(12, 1) = C(12, 2) = 0, C(12, 3) = 1. According to do Carmo and other textbooks, if M1, M2 and M3 is the basis for the left-invariant vector fields of A, where A is a member of SO(3), we have [Mi, Mj] = C(ij, k)Mk, where Mi = A * Li. In the above case, we have [M1, M2] = M3.

But, when I put A = [cos(t) -sin(t) 0; sin(t) cos(t) 0; 0 0 1], then M1 = [0 0 sin(t); 0 0 -cos(t); 0 -1 0], M2 = [0 0 cos(t); 0 0 sin(t); -1 0 0], and M3 = [-sin(t) -cos(t) 0; cos(t) -sin(t) 0; 0 0 0]. By straightforward calculation, it can be seen that [M1, M2] is not equal to M3.
I believe the textbooks could not be wrong , but my calculation is also correct. I am in confusion. Please help me.

 

[Jhero]

I understand your confusion and I am happy to help clarify the issue. First, it is important to note that the structure constants of a Lie algebra are not unique and can vary depending on the basis chosen. In this case, the structure constants you have calculated are correct for the basis {L1, L2, L3}. However, the basis you have chosen for the left-invariant vector fields {M1, M2, M3} is not the same as the standard basis {L1, L2, L3}. This is why you are getting a different result for [M1, M2].

To see this more clearly, let's look at the definition of the left-invariant vector fields: Mi = A * Li, where A is a member of SO(3). In your case, you have chosen A = [cos(t) -sin(t) 0; sin(t) cos(t) 0; 0 0 1], which is a rotation matrix. However, the standard basis {L1, L2, L3} is not a rotation matrix. It is a basis for the Lie algebra of skew-symmetric matrices (so(3)), which is a different mathematical object than SO(3). This is why you are getting a different result for [M1, M2].

In order to get the same result as the textbooks, you would need to choose a different basis for the left-invariant vector fields that is compatible with the standard basis for so(3). For example, you could choose a basis of left-invariant vector fields that are also skew-symmetric matrices, such as {A * L1, A * L2, A * L3}. In this case, you would get the same structure constants as the textbooks: [A * L1, A * L2] = A * [L1, L2] = A * L3.

I hope this helps clarify the issue. Remember, the choice of basis can greatly affect the structure constants, so it is important to be consistent when comparing results. Keep up the good work in your studies!