I'm putting this question here because I can't get any help from the HW forum (It's actually not a HW, but it looks a lot like a HW, so I won't be surprised if it gets moved there).(adsbygoogle = window.adsbygoogle || []).push({});

Source: Anderson,Principles of Relativity Physics

p. 13, prob. 1.4

"Reparametrize the rotation group by taking, as new infinitesimal parameters, ε^{1}= ε^{23}, ε^{2}= ε^{31}, and ε^{3}= ε^{12}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 y^{i}= x^{i}+ Σ_{j}ε^{ij}x^{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:

y^{i}= x^{i}+ Σ_{k}ε^{k}f_{k}^{i}(x)

where the f_{k}^{i}(x) satisfy the following condition.

(3)

The request for structure constants is a request for constants c^{k}_{mn}such that:

y^{i}= x^{i}+ Σ_{k}Σ_{m}Σ_{n}(ε_{B}^{m}ε_{A}^{n}- ε_{A}^{m}ε_{B}^{n})c^{k}_{mn}f_{k}^{i}(x)

(4)

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

My problem with understanding:

I don't know how to find the f_{k}^{i}(x). I have:

Σ_{j}ε^{ij}x^{j}= Σ_{k}ε^{k}f_{k}^{i}(x)

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

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# Structure constant calculation

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