- #1
Kontilera
- 179
- 24
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:
[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 doesn't this mean that he will find:
[tex][X_a, X_b] = 0 \quad?[/tex]
Thanks in advance!
// Kontilera
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 doesn't this mean that he will find:
[tex][X_a, X_b] = 0 \quad?[/tex]
Thanks in advance!
// Kontilera