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I have a problem regarding coordinate and non-coordinate bases.

As I understood from my course in GR, the partial derivatives of a coordiante system always commute:

[tex][\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}] = 0 .[/tex]

Which is not necessarly true for non-coordinate bases.

However in Giorgi's book 'Lie Algebras in Particle Physics' he starts out by parameterizing a Lie group G, by a set of N real parameters. (I.e. a coordinate system.)

The he shows that if we taylor expand around the identitiy (for a representation) the we get a set of generators which are the Lie algebra of our Lie group G.

My question then is:

Since he started out by choosing his generators from a coordinate system doesnt this mean that he will find:

[tex][X_a, X_b] = 0 \quad?[/tex]

Thanks in advance!

// Kontilera