Understanding Lie Algebras: Structure Constants and Commutation Relations

[welatiger]

the commutator in the Baker-Campbell-Hausdorff formula must be proportional to some linear
combination of the generators of the group (because of closure)
The constants of proportionality are called the Structure Constants
of the group, and if they are completely known, the commutation relations between
all the generators are known, and so the entire group can be determined in any
representation you want.

I want to understand this?

 

[strangerep]

I want to understand this?

The exponential map is an elementary part of the theory of Lie groups. A Lie group depends continuously on some parameters, and differentiating a group element wrt these parameters near the group identity yields the Lie algebra -- which can then exponentiated to recover the group.

Try Greiner & Muller "QM -- Symmetries" for a physicist-friendly introduction. Or maybe even Ballentine's "QM -- A Modern Development".

[Maybe this thread belongs in the group theory forum?]