Structure constants of Lie groups

[turin]

Source: Anderson, Principles of Relativity Physics

p. 13, prob. 1.4

"Reparametrize the rotation group by taking, as new infinitesimal parameters, ε¹ = ε²³, ε² = ε³¹, and ε³ = ε¹² and calculate the structure constants for these parameters."

My assumptions:

(1)
The ε^ij mentioned in the problem are the infinitesimal Cartesian parameters of the 3-D rotation group such that ε^ij = -ε^ji, and yⁱ = xⁱ + Σ_jε^ijx^j, where x is the original point and y is the transformed point.

(2)
To generalize this to non-Cartesian coordinates and still maintain the Lie group-ness, the transformation takes the general form:

yⁱ = xⁱ + Σ_kε^kf_kⁱ(x)

where the f_kⁱ(x) satisfy the following condition.

(3)
The request for structure constants is a request for constants c^k_mn such that:

yⁱ = xⁱ + Σ_kΣ_mΣ_n(ε_B^mε_Aⁿ - ε_A^mε_Bⁿ)c^k_mnf_kⁱ(x)

(4)
The parameters ε^k are the non-Cartesian parameters, and so, they should multiply some functions f_kⁱ(x), and these functions determine the structure constants.

My problem with understanding:

I don't know how to find the f_kⁱ(x). I have:

Σ_jε^ijx^j = Σ_kε^kf_kⁱ(x)

but I don't see how this tells me f_i^k(x). Am I supposed to assume some kind of orthogonality or something?

 

[Nate810]

Do I need to use some sort of change of coordinates? If so, how? Can anyone help me out? Thanks!

 

[M98Ranger]

Thank you for your question. The structure constants of Lie groups are important in understanding the algebraic properties of these groups, and your question about finding the fki(x) is a valid one. Allow me to provide some clarification and guidance on how to approach this problem.

Firstly, your assumptions (1) and (2) are correct. The infinitesimal parameters εij are the generators of the rotation group, and the transformation takes the general form yi = xi + Σkεkfki(x), where fki(x) are functions that satisfy certain conditions to maintain the Lie group structure.

To find the structure constants ckmn, we need to consider the transformation of a point x under the new parameters ε1, ε2, and ε3. Using the given reparametrization, we have:

yi = xi + ε23f1(x) + ε31f2(x) + ε12f3(x)

Expanding this out, we get:

yi = xi + ε2(f1(x) - f3(x)) + ε3(f1(x) + f2(x)) + ε1(f2(x) - f3(x))

Comparing this with the general form of the transformation, we can see that:

f12(x) = f1(x) - f3(x)
f13(x) = f1(x) + f2(x)
f23(x) = f2(x) - f3(x)

Now, using the condition mentioned in assumption (3), we can write:

f12(x) = -ε3f13(x) + ε2f23(x)
f13(x) = ε3f12(x) - ε1f23(x)
f23(x) = ε1f13(x) - ε2f12(x)

Substituting these expressions into the general form of the transformation, we get:

yi = xi + ε1f12(x) + ε2f13(x) + ε3f23(x)

Comparing this with the given form of the transformation, we can see that:

f12(x) = f1(x) - f3(x) = f1(x) + ε23f3(x) - ε31f2(x)
f13(x) = f1(x) + f2(x) = f1(x) - ε23f2(x) + ε12f3(x)
f23(x) = f2(x) - f3