I'm studying representation theory in order to have a basis to study quantum field theory.
I think the text (my professor's) i'm studying on is pretty confusing. I don't really get the difference between the representation of a group and its adjoint representation.
What I understood is that the A. Representation consists of nxn matrices, where n is the group dimension. But I thought this was also for any representation of the group, because on this text they are always written as matrices.
Thank you very much
The adjoint representation is an example of a representation of a group, but it's not the only representation; it is one of many. In fact, a common thing to do with Lie groups is to classify all possible representations (or at least the irreducible ones, up to isomorphism).
For example, the adjoint representation of $SL_2$ is made out of $3 \times 3$ matrices. However, we can make a representation (in fact, an irreducible representation) of this group with $n \times n$ matrices for any $n$.