How to get from representations to finite or infinitesimal transformations?

[confusio]

Hi all. I have here a reference with a representation of the Lie algebra of my symmetry group in terms the fields in my Lagrangian. In order to calculate Noether currents, I would like to use this representation to derive formulae for the infinitesimal forms of the symmetry transformations described by the elements of the Lie algebra. I know how to get from the finite forms of the transformations to the infinitesimal ones, so those would be great too. Does anybody know how to get from the representation of the algebra to the actual transformations?

Thanks so much for reading.

 

[tom.stoer]

... with a representation of the Lie algebra of my symmetry group in terms the fields in my Lagrangian. ... Does anybody know how to get from the representation of the algebra to the actual transformations?

As far as I understand it you have fields like A^a(x) where a is the algebra-index. Now you want to find the transformation from A(x) to A'(x), correct?

So what you need is the finite transformation. They can be represented via the matrix representation. So you have to find generators of you algebra, i.e. matrices T^a which form a basis of the algebra in some representation (fermions usually in the fundamental rep., bosons like photons, gluons, ... usually in the adjoint rep.). The group-valued transformations are generated as

g[q] = exp iq^aT^a with q^a=q^a(x)