# Regarding coordinate and non-coordinate bases

1. Oct 7, 2013

### Kontilera

Hello PF members!
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:
$$[\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}] = 0 .$$
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:
$$[X_a, X_b] = 0 \quad?$$

// Kontilera

2. Oct 7, 2013

### lavinia

Not having read the book I am not sure but usually with Lie groups one looks at the Lie bracket of left invariant vector fields

3. Oct 7, 2013

### dextercioby

Georgi's book is not a tool to learn mathematics. Moreover, if you have a diff. geom. background, by propery going to Lie groups + Lie algebras would soon make you realize that most of what's written in physics book is wrong or at best confusing. My advice is to capture the Lie groups and Lie algebras theory from a more specialized (math-oriented) textbook such as Barut and Raczka.